"Pseudo-medieval" tells you that they don't have universal health care.
Psuedo-Medieval tells you nothing of the sort. It tells you that it is "fake, spurious, sham" Medieval. It tells you only that it will have some Medieval trappings but depart significantly from actual Medieval in many important ways. Most notably, since "Psuedo-Medieval" generally involves magic, and that magic can cause healing, there is reason to expect that the level of healing available might differ markedly from actual medieval.
In practice, very little ever published for D&D then or now is actually Medieval in setting, and even not much is compelling pseudo-Medieval beyond the weaponry - and even it is biased toward 15th and 16th century arms, usually but not always, sans gunpowder. Much of the urban setting could be equally regarded as "Psuedo-Dickensonian", and could just as easily be the setting of Twain's "Peasant & The Pauper" or Dicken's "Oliver Twist". Much of the technology presented in D&D, and certainly D&D from the early 1980's on, tended to be Early Modern mixed with a wide variety of anachronistic settings from Ancient Greece to Victorian England. Feudalism, serfdom, shortages of coin, and so forth rarely are important to D&D settings or stories. Homes are generally not wattle and daub construction single room affairs in the published texts. Cities are generally much larger than their medieval counterparts, as are the armies that they can send forth. Nations tend to be monocratic consolidated nation states, not feudal kingdoms. Trade tends to be widespread, and communities tend to be cosmopolitan.
Of course, you are entirely free to adapt D&D to a more rigorously medieval game if you wish, but there is a not a lot of evidence that the publishers of the game ever thought this a particularly important thing to do. Gygax was at least as much of an Egyptophile as he was an aficionado of medieval life, and by his own accounts was just as inspired by Westerns and the Klondike gold rush as say, the Ottoian dynasty.
In medieval times life for a peasant was nasty, brutish and short. And it wasn't that much better for the nobility.
Surely you aren't going to start telling me about Medieval History. I can swing around a Charles Oman text just as well as you likely can.
Sure, you could set your adventure in a utopian society, but that would be very different to a typical D&D setting.
Yes, but you've yet to establish that a "utopian society" is a necessary or even likely outcome of all clergy presented in a setting being clerics.
A typical medieval village would have around 50 people and one priest (and if you want to assume "most villages don't have a priest"
Oh swell. You are going to try to tell me about medieval history.
There is no reason to suppose your hypothetical cleric serving your hypothetical hamlet is more than 1st level. As such, there is no reason to suppose that the level of healing involved, while it would be significantly better than that available to a medieval peasant, would lead to a "utopian" society. Injuries that might otherwise take a laborer from their employment for days or weeks could in fact be healed quickly, and minor wounds could be healed before they became septic or infected. This would surely improve life in the community, which could go a long ways toward explaining the greater prosperity typically seen in D&D pseudo-Medieval worlds than in actual Medieval settings. But conversely, his cleric cannot cure disease, and must contend with equally active forces of evil. Nor can he restore lost fingers or toes, bind broken bones, or heal damaged minds after concussions or other all too common serious injuries that occur in a rural setting with a lot of sharp objects being employed regularly. For these more serious injuries, a pilgrimage to some mightier temple would be required.
It seems the entire basis of your "proof" is your conception regarding what the game is supposed to be like, and not in fact what we can draw from published examples, demographics, or inference. You have a preferred notion and a preferred take on things ("Greyhawk is supposed to be grimdark."(?!?!?)) and by golly, you are going to stick with that, facts and evidence be darned.