Just to get briefly involved here... *okay, sorry, no so briefly.*
It would be pretty silly to believe that 1) people would choose to take time our of their lives to go sit in a waiting room for a doctor who's just going to tell them "Oh, that's it? You can pick up some OTC ibuprofen at a drug store." Boom, next person. It just won't happen; people have better things to do. Besides, there's a massive difference between a clinic and an ER... and 2) given that hospitals already prioritize, treating the more grave situations with more urgency even if others have been waiting for hours, why would providing health care make them any less organized? No hospital is going to say "well, we know you're dying due to your open stab wound, but there were just sooo many people here before you and it just wouldn't be fair to them, you know? Not after this free health care thing took off, at least." And if they do start doing that, well I guess I'm just a dumb asshole. haha
As for the economic issue, if we had to start treating everyone then I'd agree that the cost is too high. However, it's doesn't account for a couple things. The first one is that Americans are already paying for other people's extremely expensive health bills. When a dude goes into a ER, he already gets shit service and when he can't pay his bills, the burden is passed onto taxpayers anyway. Second, we currently place no emphasis on preventative care. We make absolutely no effort catch illnesses early when they're treatable, and when we do, there's nothing anyone will do about it (unless you've got the cash). Instead, we wait until you're litarlly dying, let ERs clog up, and end up spending significantly more than we ever would have if we'd simply taken measures to find and treat the thing. At the beginning, yes, taxes will be a little higher (this isn't even counting the very rich who don't pay any), but that's a perfectly reasonable exchange for health security, which a lack of is one of the few things that can destroy entire families. However, costs will down and we'll eventually be paying much less than we are now.
As for the quality, probably won't change. A doctor charges $1000, he gets $1000. Doesn't matter if it comes from you, your insurance company, or the gov't. UHC doesn't turn hospitals into the DMV; they remain their "independent" selves. In fact, if you get in an accident and you're being taken to a hospital, you hope they take you to the right one. To give a story 'bout my high school econ teacher, haha, guy got into a motorcycle accident one day. He was messed up pretty bad, ended up losing an eye, and when he got the bill he found out his insurance wouldn't cover it. Of course, he demanded to know why, and apparent the ambulance that picked him up took him to the closest hospital... one that happened to not be covered under his particular company's plan. haha They fixed him up okay, not any better or worse than the other hospital would've, but he ended having to pay the whole thing himself. I can't say I know what's goin on in France, but I do know that there are many ways to handle health care. You can regulate insurance companies, make a gov't insurance program, regulate pharmacutical companies, give EBT-like vouchers, set up public hospitals, and so on. Standards don't automatically disappear. The US has some of the best health care in the world; it just so happens that no one can afford it and insurance companies exploit that fact.
Furthermore, you can't really compare the US to France in that regard. Yes, we have more people, but we also have almost a 6x greater GDP. There's no reason to think more people will need treatment either. In fact, significantly less people will need treatment. People will be able to go to clinics and get themselves checked out. No reason to believe it will result in crowded clinic either, because as the country moves toward an emphasis of preventative care, a percent of the health care spending will go to building more clinics and hospitals. We don't have that now, because of have the backwards mentality to wait for the ER when the problem can be addressed much more efficiently and cost effectively with preventative care at a clinic. And if it did cost more (which I agree it will in the beginning due to that economic law I don't remember the name of), it doesn't mean spending cuts. It's far more likely that it would result in a tax increase, meaning the quality doesn't change.