Most of the cats get fixed and then put out into the barn. Their lives are fairly short usually but we have a few that are tough scrappers.
Dogs though... when you have your quota what do you do with them? If you leave them to run loose then you end up with a pack which you then have to hunt down and kill. Hopefully before they kill someone.
Unless there is something special about the dog they are put down. It is a crying shame because some are very nice dogs.
We don’t get many dogs either but when it happens it sucks. Cats I can deal with but some of these dogs( read, most) were pets and many ware well trained. We’ve let them roam mostly and some adapt. Cats are out of control. We are far enough away from town it’s a perfect dumping ground and yet close enough for daddy or some evil sh&t to dump litters where other cats are and there is no answer except kill them. I’m no longer paying to neuter, nor putting up signs. I;m done with that. Happened the first couple years and it was enough and costly.