Except it isn't an argument against nationalised healthcare. The UK has nationalised health care and that is what OP is complaining about. I think it would be more an argument by OP against "nationalised healthcare with legal private healthcare".

The exact argument is "doctors/nurses are limited, and they aren't behaving how I think they should behave. Let's fix the system so they have no choice". I guess that isn't slavery, that is just funneling people into a single path which they clearly don't like, because they reject that path when they have literally more than one option open to them.

Who is doing the rejecting and regarding what path? What specific proposed fixes are you referring to?

It seemed like OP was arguing that private medical practices, which are still allowed in England, should be banned because they're using up all the "resources" the NHS needs (doctors/nurses).

I agree that a ban is ineffective. In lieu of a ban the NHS should raise their staffer's pay and fund the increase with financial asset and land taxes.