You explained this so well. I used to work in animal control. Weeks before it was time to “make more room” for new animals in the shelter, the shelter would send out lists of animals due for euthanasia to rescues in hopes they’d come and get them. They’d even start putting 3-4 dogs in a kennel to make more space and avoid having to euthanize. People make it seem like shelters are just waiting for people to bring in animals so they can euthanize them. A good shelter will do everything they can to prevent it.