This works because the actual service is not monopolised by the state but rather a large pool of providers.
This doesn't apply to other services.
And public schools. And public communication infra. The list just goes on and on. It's not 'just healthcare' and the amount of data on this is so staggering that to suggest that the private sector running things somehow results in an efficiency gain of some kind is borderline brainwashed rhetoric from Fox News more than it is any kind of defensible position.
Do you have any numbers to support this? Governments spending stupidly higher than private sector for this, I know this as a matter of fact.
Same for the other claims about "public schools. And public communication infra." Per unit of service provided, governments spend upwards of 2 to 5x in major western countries.