These good citizens also post to their locally-run "lost and found" Facebook pages. There's a whole online network dedicated to helping lost pets. If people use such methods, a physical shelter is unnecessary.
Now, I'm not saying that animal control shouldn't be a thing. It totally should. If a dog is in danger or a danger to someone else, they should be called. But good luck with that, because in my home town, as far as I know, it's not animal control that comes out. At best, it's the police. However, there should be no physical shelter that the animal control officers take the animal to. The animal can be removed and placed somewhere else where it is safe. Or, if it is rabid, it can be euthanized on the spot.
At this point in the conversation, my father cried, "But this would only lead to people shooting dogs!" As if that doesn't already happen in many places (and just for fun). And yes, it might even increase dog shootings, but at least our tax dollars wouldn't be paying for it. I, personally, would not be financially contributing to the death of millions of animals a year.