In my mind, it's not charity because it's
- For everyone. Maybe not for you today, but maybe tomorrow
- For the benefit of society as a whole. If we let everyone who got sick just die, our society would be worse for it.
If you are thinking of something that is for literally anyone, you are most likely thinking of universal healthcare or some other like system.
We've seen how the judgement of those receiving aid has affected US policy in the past - a handful of people are marked out as undeserving of the aid (maybe someone with a long term physical disability commits murder or something else) and then the full category of people are painted with suspicion (ala "welfare queens") and that suspicion is used as a justification to cut spending drastically under the guise of oversight. There are a fair number of people clamoring today for UBI for the reason, in my view at least, that it's the only way to undo all the complicated conditions on various forms of governmental aid.
I'd personally be quite in favor of keeping it dry, cold and governmental - healthcare is, IMO, an important enough topic that we can't let emotions override treatment. If we're going to put a bunch of money into a big pot we should have some very deliberate rules about how that money comes out and what it gets used for and posing it as a charity is going to get all sorts of rules applied to it (i.e. it can't be used for anything Planned Parenthood related, it can't be used for contraception, it can't be used by trans identifying individuals, you need to have a full time job to access it, it is unavailable to felons, etc...) - some of those rules might make sense, others might not. At the end of the day I think posing it as a charity will make it a lot harder to come up with a fair set of restrictions on it.
I think it's more a question of whether we'd be willing to install and maintain a roof on a building that had been designed without a roof in mind or whether we'd tear down the building to make room for a new one. You'd probably rationally lean in the second direction for a house - but when we're talking about a human life the math changes for a lot of people.
Either way, a return to an insurance-less situation would leave people with preexisting situations out in the cold, unless your definition of insuranceless involves government subsidies for all preexisting conditions and at that point you're basically talking about medicare for all.