kinda?

it's more the sense that evne when you are providing a free option there tends to be caveats to it that some people wont be interested in(make no mistake "Free healthcare" is not free, in order for this system to function properly thre needs to be regulations put in place in the interest of protecting the Health of the population, and this is where you have a lot of pushback from lobbies un the US).

the common complain with NHS performance is wait times and the perceived notion that the quality of care is gonna be lesser(this is not true), so they might sitll want or need access ot the private sector, this is where having Health insurance is gonna matter and where the demand would remain.

but ultimately the big shake up would be, that with a free altenrative available, Health insurance providers would have a ceiling on how much much they can charge clients and hospitals(since they have a choice now)

this is also to mention that many jobs that were placed in the private sector, would just get transffered over to be managed in the public sector(you still need people to coordinate hospitals in order to assure that payments and the financial side of running them are taken care of.)