What does this mean? You could argue our current system is exactly this. People can get healthy relatively easily, if they’re relatively wealthy. You could say instead “All people have a right to healthcare” which implies that anything standing in the way of that is at its core unjust and immoral.

I'm losing my mind. I feel like this entire website has Natural Rights Theory brainworms. His formulation is not great, but it is clearly superior to yours.

If being wealthy is a necessary element to being healthy in a given society, then in any such society where it is also the case that social mobility even vaguely resembles ours conceptually (or is much worse, like in a caste system), it logically follows that not everyone can easily be healthy in such a society because the existence of a populous, downtrodden underclass is a necessity of that system and therefore lack of access to health resources is a necessity of that system.

His, uh, axiom, wish, whatever you want to call it is worded in an overly soft and potentially game-able way, but talking about "rights" independent of material reality is completely useless.

Now, if you mean a legal right to free healthcare, that's something else, but the rest of your statement implies a moral prescription which, by itself, is worthless.