Whether or not people want the government to do it implies nothing about whether the government would be good at doing it. And neither of those things imply anything about whether private insurers are good at checking doctor's orders.

I would say that for any reasonable definition of "good", whether in skill or virtue, private insurers have shown they are definitely not good actors in our health system. Our health system might just be better off without them.

As for your question, I for one would like _doctors_ to double-check doctors' orders. Not an actuary.