Reading into your intention, this is actually more like health insurance than single payer healthcare. Not quite a million little coops, more like a few dozen. And it would end up having most of the same problems of modern US health insurance.

You'll need someone to administer the program, so you have to give them some power over your money. That means they'd need the power to say "no" to people who are seeking healthcare resources for invalid reasons--things like Munchausen's syndrome at first, but eventually they'd have to make calls about things that people actually need but can't prove they need, just like health insurance does now.

If you don't want do these things, I guarantee your neighbors will insist they be done (ever hung out on nextdoor? those are the people you'll be pooling your money with). And you'll go along, because it's a hassle not to, and hey at least you're getting your needs taken care of most of the time. If you manage to keep your program free of capitalist influences, you're going to have to fight corruption instead: "Slip me some dough and I'll make sure you get seen next."

So in time you just end up with health insurance, and most of its flaws, if you don't very carefully watch the people administering your program, if you don't very carefully fight against the perverse incentives.

The biggest problem, of course, is that existing health insurance would fight it like penicillin fights bacteria. They have had decades to do regulatory capture in their benefit, and if another group comes along that's almost-but-not-quite health insurance, they're going to make sure that the regulations they captured keep it from going anywhere, up to the point of trying to make it explicitly illegal.

I think we're in agreement about single payer, but this ^ is how it benefits us. The government has actual power to fight corruption and isn't beholden to capital. Now if we only had a way to create a just government.