what you suggest simply will never be an effective strategy.
Doctors can also be incentivized to do rotations in the ER. My primary care physician many years ago used to do ER rotations as a sort of overtime. Inventivize with the right pay, more will do it. Granted this isn't a large labor supply, but it certainly could help.
Instead, what you see is a shrinking of the nurse labor supply by a few percentage points due to vaccination requirements. Pretty ridiculous to complain about lack of capacity when you force some of that because you refuse to officially recognize natural immunity as equal to vaccine derived immunity.
nurses and doctors are leaving due to the stress of a long term pandemic. are you asserting we can just force them back into labor? that's fairly dictatorial of you. it still doesn't change that we would only revert to original levels, it would not be a significant increase in supply.
'incentives' won't help. there simply isn't enough supply in aggregate. and they are not leaving because of a lack of pay they're leaving due to the stress of dealing with assholes on the daily. cash isn't going to fix that problem.
the only option left in such a scenario is literally mandating they return to work. which doesn't nothing be return you to initial workforce levels it doesn't increase capacity.
we're seeing this right now with the national guard 'mobilization' in some areas. they're literally doing nothing but sitting with patients and other menial tasks that don't require training. they're basically just in the way.
plenty of people in the 100-200k range take cuts in the 30-40k / year range to move to better work environments. so the idea you'll get doctors who generally make 200k+ would be willing to work in shitty conditions for a prolonged period is pretty bogus.
nurses you might be able to entice them to come back for 20-30k bumps, but its also unlikely.
even if your idea worked, it doesn't increase the supply. it just brings us back to predecline levels.
edit: in fact a clear example of your idea not working: paying people to get the vaccine. did jack shit to actual increase uptake.