Hubbard faced a lot of obstacles when it came to writing a good story. He was trying to write science fiction without understanding science, and trying to write about people when all the evidence suggests that he only fully comprehended the most negative of emotions and motivations. His plot twists were never really clever, and if he had an original idea, chances were it was something like monsters who explode around radioactive particles, or Teenie.
That said, his premises aren't entirely without merit, and I've already mused how an editor or more competent writer could salvage even things like Mission Earth. So let's wheel Ole Doc Methuselah into the operating room and see how we could keep him out of the morgue. Which is a metaphor that falls apart when the author names the hero-doctor's spaceship the Morgue. Hrm.
Anyway, isn't it strange what little effect Ole Doc being Ole Doc has on these stories? The key component of this character - or at least his name - is that he's centuries old, but that never really seems to matter. When he's doctoring he's more reliant on super-advanced gizmos and magical healing salves to cure patients than his long medical career. The exception would be "Plague" when he's able to identify a mysterious illness because one of his first cases as a doctor was to help a woman with measles, but that wouldn't have been necessary if his Universal Medical Society had kept better records. I suppose one could argue that Ole Doc's age helps to sell the feats of marksmanship or other action hero stuff we see him perform, except 1) that begs the question of why he needs to be a doctor in the first place and 2) we've seen similar stunts from non-immortal Hubbard protagonists. Frankly, we could swap Ole Doc out for a regular physician with access to the same medical technology with no real effect on how the stories unfold.
So that's the first thing to change: make Ole Doc's age actually significant beyond how it lets him remember that measles is a thing. Now as I said, any young physician could play Ole Doc's part in these stories - that's the problem with having a way of storing information in books or computers, it lets people access hundreds of years' worth of knowledge without having to be hundreds of years old themselves. So instead of having Ole Doc's age impact the plot, how about make it impact his character?
A competent author could use an unnaturally-old character to explore what they think the effect of hundreds of years of continued living would do to a person's mind. Now, you could play this for a lot of drama, make Ole Doc a grim figure buckling under the weight of centuries, someone who has watched the same problems and tragedies unfold again and again, someone who has seen everyone he ever cared for crumble into dust, someone whose mind is straining to contain more experiences than a mortal man is meant to endure. But that's not really the kind of light-hearted fiction Hubbard was originally going for.
So instead you could make Ole Doc a bit scatter-brained, someone who can lose track of what day it is, or what century it is. For someone who has met so many people and been to so many worlds, proper names become unimportant for him: he never refers to other characters by anything but their title or occupation, and doesn't bother to remember a planet's name or what its capital is called.
"But he already does some of that," you might point out. And that's true, Ole Doc can be forgetful and has trouble getting along with people who don't have a great ass. But Hubbard has Ole Doc forget to do important things like take his life-extending medicine and fuel up his spaceship, which leads the reader to question how Ole Doc survived so long in the first place, and doesn't make the connection between Ole Doc's treatment of other people and his extreme age. His Ole Doc doesn't come across as someone trying to handle several lifetimes' worth of experience, he comes across as an improbably-surviving asshole.
The key is to give Ole Doc some characterization that shows why he's continued to do medicine for hundreds of years. So it wouldn't hurt to make him interested in his fellow humans beyond the Girl of the Week. He may not be able to remember his patient's name while he's working on them, and probably won't remember them once he gets on his spaceship and leaves, but have Ole Doc engaged with them, getting along with them, interested in who they are and how they're doing. Show that their lives matter to him. That slavery in general bothers him, not just the enslavement of a would-be supermodel in particular. That if a horde of babies are about to be born on a planet with one caretaker, Ole Doc isn't one to walk away from the problem to go fishing. Show that he's a physician who will teach people to take better care of themselves, not a technician who swoops in with a pharmaceutical ray rod when things get dire and leaves until there's another great emergency. Or he feels like going fishing.
However, making Ole Doc sympathetic and altruistic crashes headlong into the backstory for Hubbard's Ole Doc stories, the whole Universal Medical Society and its medical monopoly. If you really want to keep this angle - that there's an elite organization that has decided it is the only group moral enough to properly use advanced medical knowledge, but won't use that knowledge on anything but the greatest medical catastrophes; a group that it is so powerful that it can boss around planetary rulers with impunity, but so dispassionate that it doesn't care if those rulers are murderous tyrants because it doesn't want to get involved in "political" matters - well, this is a premise that would work best in some dystopian fiction, not an allegedly positive pulp story.
Of course, removing this Universal Medical Society does present a problem in itself. If Ole Doc's doodads and textbooks become available to the masses, what makes him special? Besides the age thing. And this is probably why Hubbard came up with the UMS in the first place - since he's not doing anything with Ole Doc's age, and doesn't know enough about medicine to write Ole Doc as a super-doctor, the best he can do to make his hero noteworthy is make everyone else in the galaxy a moron. The sad fact of the matter is that a galaxy without the Universal Medical Society doesn't need those Soldiers of Light to protect it.
The solution to this problem is to do what Hubbard doesn't and make something out of Ole Doc's age. So in a UMS-less galaxy, where every local physician has access to ray rods and medical encyclopedias, Ole Doc's advantage is that he doesn't have to spend hours scrolling through computer screens or paging through tomes to know what disease he's looking at. He may not have a perfect memory, but when he sees someone turning colors and frothing at the mouth, he'll remember the common symptoms of Algolian Flu and brew up a fix in a jiffy. He many not be able to remember names, but hundreds of years of interacting with patients have given him the people skills to set others at ease in even the worst crises and coordinate a response when everyone's first inclination is to panic. He may not always be up to date on the newest ray rod model, but he knows how to treat an illness without one, allowing him to act during power outages or in primitive hospitals when his fellow doctors would be helpless.
This talk about Ole Doc's age brings up another issue with these stories, or rather a question: why is Ole Doc one of the few immortals in the universe? I'm not asking the author to explain the formula or whatever, I'm asking why Ole Doc has kept this revolutionary discovery to himself, even before the formation of the Universal Medical Society and its restrictions on what lesser men are allowed to know about medicine. Does he have some philosophical reason for keeping it secret, like he thinks society couldn't handle a deathless population? Does the treatment come with a terrible price, one that he thinks is only justified if he uses his extended existence to save others' lives? Does he even remember how he came up with the immortality elixir, or was it a happy accident? Is he still working out the kinks in the formula, still monitoring himself for any potential side-effects until he feels like it's safe to share with everyone else? Or has he come to hate his condition and is contemptuous of any fools who would wish to join his eternal existence?
A good course of action would be to have the life-extending treatment be part of becoming a Soldier of Light - you get to live potentially forever, but sworn to the service of your fellow man and a life of constant toil. I suspect this is what Hubbard was going for, but again he never bothers to explain it, and the notion is at odds with all the times we see Ole Doc loafing around or refusing to get involved in some problem or acting on whimsy. And since I'm against the idea of the Universal Medical Society to begin with, I'd say it's better to have Ole Doc's agelessness be something unique to him and unable to be replicated. That way we don't have to explain why angry mobs aren't constantly trying to dissect him and figure out what parts they need to eat to gain his immortality power. Heck, maybe it'd be best if nobody knows Ole Doc is ageless, and he's just some odd drifter who comes in on an old spaceship but turns out to know a ton about medicine.
Only that wouldn't allow Hubbard to shower a protagonist with fame and fortune and gilded luxuries, and by now you know how important that rot is to him.
Finally, there's the matter of Hippocrates. We know that Ole Doc purchased him at an auction as a medical curiosity, and we know that the alien is probably the only thing keeping Old Doc alive. We can guess, since Hubbard doesn't properly develop the notion, that Ole Doc likes Hippocrates because the alien is almost as long-lived as he is, so the gypsum lizard-thing serves as a reassuring constant in an always-changing universe. And we know that Hippocrates is incredibly devoted to his "owner," but we never learn why. It's only in "the Great Air Monopoly" that Ole Doc shows any real concern or affection for his "slave," so Hippocrates' willingness to put up with Ole Doc's assholery is just plain perplexing. And then there's Hippocrates' purpose within the narrative: in the stories as written, he's a combination butler and comedic relief character, someone to fuss over Ole Doc and make sure he eats his life-extending miracle drugs, but also whine and recite regulations for hours because that's humorous or something... I didn't say Hippocrates is an effective comedic relief character.
If it were me, I'd drop the "slave" and "butler" angle entirely, and let Ole Doc know how to take care of himself. Instead, Hippocrates could be a fellow doctor, someone from a similarly long-lived species who wants to see how Ole Doc treats his patients. And I don't mean Hippocrates wants to learn how to fix a human's broken arm - we can let him keep his perfect recall and have him read all Ole Doc's medical texts for that. Instead, Hippocrates could be trying to learn about how to deal with patients, how to make some of the hard choices a physician is expected to handle in a crisis. You could contrast the alien's stuffy, by-the-book approach and lack of diplomacy with Ole Doc's centuries of experience and good ol' country boy heart and soul. And to keep the exchange from being one-way, maybe Ole Doc could learn things from Hippocrates, like how to use the latest medical gadgets and what's going on in the galaxy. Since Hippocrates is naturally long-lived, maybe Ole Doc hopes to learn how to handle his own extended lifespan.
So, there's some ideas. All you'd have to do to fix the "Ole Doc Methuselah" stories is overhaul the two main characters and the tales' basic premise. And then decide what to do in those stories, since a non-stupid galaxy wouldn't let an author get away with plots like "the bad guys were using ragweed to sell 'air'" or "everyone else forgot what measles was" or "they forgot to check for radiation when visiting a strange planet." Or in other words, more or less completely start over from the idea of "he's a doctor and immortal and in space."
Alternatively, you could decide that's too much effort and read a better book about an immortal protagonist. For example, I think Robert A. Heinlein did some stuff with a fellow named Lazarus Long, who... wow, his mom and his twin opposite-sex clones?