german, currently living in the US here: the scenario OP describes would unfortunately be the same in germany. The healthcare is good, but slow. And any non-obvious diagnosis can take a while / several specialists to figure out.

Dunno if the private health care would be any better (wait time aside), but if anything I'm surprised you can't just throw money at the problem while in the US. Did something similar to what you described twice recently and for each condition it took less than 2-3 weeks between GP, specialist and results.

You don't even have to move I think you can likely get treatment through a private hospital on a tourist visa.

"Excellent free" might still get OP to the same delay loop he tried to escape. His US doctor might work "for the insurer" but the doctor in the free health care system might equally feel they are working "for the state". In my experience the state provided health care is mostly excellent for minors except for psychiatry which is underresourced, but for adults its roll of the dice (outcomes ranging again from excellent to abysmal) with long queues for specialists (Finland).

With wait times triple than what OP talked about..

Are there public systems in Europe where specialist waits are multiples of what OP is already experiencing?