I don't know about the USA but in the UK if you find a stray animal the vets won't take it in, you have to take it to a police station,

The vets have many cages in well kept rooms or enclosures and also proper dog food whereas the police station has a couple of scruffy looking 'kennels' and either no dog food or some cheap mixer.
Obviously if the police station has a dog section then they would be better prepared but small town stations don't.

Also if taken to the vets then they can have a quick health check and any problems dealt with there and then before moving on to a shelter but with the police they have to wait untill a shelter can be contacted and arrange for a pick up of the animal, then the shelter has to arrange for a vet check up.

Surely the whole process would be quicker easier if the strays went to a vet first.

Why don't vets take strays?