Based on my experiences (I lived in Europe for a bit), I was under the impression that our care is of higher quality. Do you have any sources for this?

The reason for this is that you're likely to only be thinking about "high effort, high pay-off" health care: Cancer treatment, emergency medicine, etc. This is only the tip of the medicine ice berg. I am assuming you're young and have not yet had the pleasure of "small effort, small pay-off" health care: Allergy medicine, post-birth care, backpain treatment, whatdoIknow.

Indeed, the US has some of the best health care providers for high effort (and costly), high payoff care, and that's a good thing! And while that should definitely stay (and can, in "socialized health care", just look at Switzerland for example), the ability to, say, separate Siamese twins is not what defines the quality of a health care system.

In fact, concentrating on high cost treatments is likely disadvantageous for that. The newest treatment is not always the most effective. If a new medicine costs five times more, and has a 1% higher success rate, is it worth it? Should it be employed to everyone, or just to patients where it might raise the success rate by more? Such high-effort, low payoff treatments are one of the drivers of health care cost while not really contributing that much to quality.

The problem is that in the US, health-care providers, by getting a cut from medicine sales, being able to offer more high-effort low-payoff helps their bottom line: It simply puts more work and money into a provider's bank account. Doctors and hospitals have an incentive to offer such treatments, especially when insurances set no limits on the treatment and when some patients pay out of pocket.

In a more tightly regulated system, this incentive is not as strong - if you're doctor's salary is fixed, there is no incentive for the doctor to choose high-effort low-payoff treatments over low-effort ones.

Which leads us to your second point:

Countries with a nationalized system tend to use simpler, cheaper, older techniques (which are past the patenting threshold) a LOT MORE than we do here.

That is not a vice, it is a virtue (depending on the circumstances). You could, theoretically, give everyone who goes to the doctor for the flu a blood test to check for other possible reasons for the symptoms. That's another 100$ spent on a flu case, an often very simple case where people just go to see a professional to be able to take sick time-off from work. A simpler, more effective option (for both the system and the patient) would be to screen for risk cases for other illnesses and give the vast majority simply the medicine they really need.

Which leads me to

If we end up going with a nationalized system, I think we'll have the same thing, here. The state will mandate cheaper solutions, which means the new, emerging stuff will be less profitable for business and R&D will slow down.

That is just plain wrong. Advances in medicine are driven by a) foundational research usually paid for by government funds at public universities, b) treatment tests paid for by either government grants or private manufacturers, and c) young medical professionals interested in developping new techniques.

A) and c) are independent of the health-care system. B) is somewhat dependent, but while in Europe, medicine prices are more tightly regulated, consider that the decision-making bodies take into account R&D costs. In the US, there is no such thing for people who are not on medicare or medicaid, and that means that hospitals just aks for fantasy prices when they can, making everything more expensive.

That and the free-rider problem: In the US, there are comparably little consequences if you just go to the ER and never pay the bill. Sure, could go to collections, but the hospitals give up on getting the money after a while, meaning that the other patients have to pay for that treatment, too. In a mandatory insurance or single payer system, you don't have this problem.