Let me use an analogy:
Let's say I think birds are annoying because they eat my blueberries and crap on my windows as they fly by. Let's say I want to get rid of all birds that come onto my property. (a) Should I be allowed to kill every bird that comes on my property, simply because I find them annoying? (b) If I am allowed to kill or remove them, and take them to the local pound, should taxpayers pay for killing them for me, simply because I don't like them? Or should I pay for killing them?
What if we are talking about Deer? Raccoons? Butterflies? Worms? Coyotes? Cats? Children?
Depending on the species, I'm sure each of our answers is different.
I didn’t mean to imply that charging to drop off animals was a problem for me.
But lets be practical. If I am surrendering my own pet, by my own decisions - then I have zero problem with paying a surrender fee (though many people would just turn the animal loose/dump it somewhere). I do realize there are costs involved.
But if someone else’s pet or a ferrel pet is coming onto my property and causing trouble - why should I be forced to pay the pound to take the animal?
And your proverbial annoying birds are wild animals that animal control has no jurisdiction over. Nor does the local pound take any wild animals. Period.
So your comparison falls rather flat.
There are mechanisms to deal with nuisance wildlife here.