why would the money coming from one pocket incentivize economic optimal treatment? By what mechanism? Why wouldn't hospitals try to fleece the government (spoiler: they do, in the countries with public health care)?
In the UK most hospitals are owned and run by the state.
Single payer works better, that's why almost all countries use it for basic healthcare.
"Only one issue per visit" so they can charge for two visits. Making me visit to hear the negative results of a standard STD panel, again, just so they can charge for two visits. They charge for an extra visit just to renew prescriptions with no additional examination. Etc.
Also there are different structures for payment, insurance and pay per visit are simply not necessary with single payer though some systems use an amalgam of public/private.
(A) which countries do you mean here? Be specific
(B) Do you think hospitals aren't trying to fleece insurers and customers all the time? e.g. with opaque billing practices?