Is there any reason to believe less healthcare professionals would be employed under a universal system? Insurance claims adjusters and call centre scripted finance deciders, sure. But actual health delivery agents?
But you will lose endless billing departments at general/specialist practices, hospitals etc, because actual consumers would not engage. Hospital administrations will be driven by the government setting reimbursement rates for services that are transparent, so the argy-bargy between someone paying cash, someone getting insurance etc will disappear (mostly).
I'm in Australia, and the billing departments, even at private hospitals are minimal. Ditto at general practices. Reimbursements for GP/specialists that are not "bulk billed" (ie paid at the government rate) are automated into a nominated bank account and are usually instant.
Medicare for All in the US would move the 2/3rds that are not covered by Medicare, VA, etc into the same systems, with the reduction in overhead from 10-20% to the 2-5% of the current government run health systems.
Allowing Medicare to negotiate both drug prices (as a large purchaser) and health services will also dramatically reduce costs, because the government has the purchasing power to drive hard bargains.
You can't just conjure up a medical school out of nowhere though. That's extremely hard and expensive. You can do it if you relax your standards. Maybe by making the "hospital" part of the school optional or something. Students won't see patients but what does it matter.
Medicare, Medicaid and Tricare are all basically universal systems for segments of the population. None work well. And all are subsidized from what people with private insurance pay. And the UHC proponents rely on a stacked deck--UHC is "better" in a system that makes 20% of the score "fairness". That is, UHC. And one of the other yardsticks is life expectancy. Sounds fine--but dig deeper. A big factor in lowering life expectancy is infant mortality. Again, sounds fine--except you see a big difference in infant mortality across the developed world that is not reasonably explained by differences in their medical systems. But you see an inverse relationship between infant mortality and stillbirths. Most of what you're really seeing is whether the doctors consider it a live birth that promptly died, or consider it a stillbirth. (And, yes, we are still an outlier--but by only 1/3 of the amount claimed.)
Show me it's better without your thumb on the scale!