The government switches external firms as soon as another firm seems to be able to do it better, that isn't true for the governments own parts. That makes the two fundamentally different, one can accumulate bloat forever the other will get renewed from time to time. Private profit seeking adds overhead though so which one is better depends on the domain we are talking about, in some cases private are better in other government are better.

There hasn't been a legal obligation to have private health insurance since the Tax Cuts and Jobs Act of 2017, which was 7 years ago.

But still, your point stands, despite the specific example no longer being true.