Teachers are generally proponents of public education as well. They still need to unionize and strike from time to time. Particularly in Republican states.
Sure, it happens, and yes, there is going to be some union pushback as there always is.
However, that doesn't mean that the doctors and nurses are going to strike to preclude any and every exercise of SP negotiating power; even if you assume an SP system won't be able to touch salaries whatsoever (which is asinine), there is still plenty of room to leverage SP negotiating power and bring down the cost of drugs, and supplies, while cancerous waste in terms of administrative expenses are all but abolished (no insurers to deal with with their litany of ridiculous/complex exemptions, exceptions, etc).
Trump got elected to build a wall. He and Paul Ryan are doing everything in their power to scrap/dismantle Obamacare, social security, Medicare, the EPA, the department of education... as we speak. They might lose the next election or two, but a lot of that damage will already be done.
And it will be reversed when they lose the next election, and they have this indelible blemish on their record.
In Canada, if the Conservatives acted fully on their true desires and say, exploited a majority to dismantle our national healthcare system, I don't think it's an exaggeration to say they might never see federal level governance again and it would deal serious lasting damage to their political brand at the least (as their replacement quickly reverses the damage immediately after, and enjoys a windfall of political esteem).
Do they have this many dirtbag Republicans everywhere else?
You'd be surprised; they generally skulk in the shadows and hide behind euphemisms, pretend at positions they don't believe in and generously employ doublespeak, but they certainly exist in force.
I'm a Democrat. I take no pharma/insurer donations. I oppose it in large part because I hate the DMV, I don't want my doctor's office to turn into one, and because of the questions, I posted in the OP.
Then, with all due respect, you're either not familiar with the impressive record of SP world wide to deliver care on par with US care or better, and at far lower cost, or you're letting disingenuous Republican/right leaning propaganda crowd out its demonstrated success globally. Moreover, let's be real about this: your hospital and doctor's office are not going to be run like your local DMV, and moreover, family clinics/doctors offices are generally privately run in SP countries anyways; they simply bill the govt instead of you.