I think that there is actually a difference between getting paid, and having the costs covered by the hospital itself through non directed donations or taxes. When a vet goes to a VA hospital the government does not send a specific check to cover his fees. The costs of providing his care are covered through a budget given to the VA hospital by the government.
When someone donates to a charity, unless its a directed donation to build a new wing or something like that, the donation is going to go into the pot to help cover all of the costs of running the hospital, including patient care.