No support.
What I want to do with my pets, I should be able to do to my pets. Besides, me deleting a few pets isn't going to affect the economy. Maybe some people have the power to actually change the rarity of a pet, but that's their choice. I don't see how it's much different from people hoarding pets. Sure, the pets are still in existence, but the hoarders are holding so many of their selected pets that they aren't trading. The said pets are getting withheld from people who may not have even one of those pets yet, and isn't the whole point of a hoard to hoard? Hoarders aren't going to trade their hoard. The hoarders could be hoarding a common rarity pet that may actually be considered uncommon or even rare should they delete their hoard (minus, say, one for collection purposes).
Some people have said "then just gift those pets" or something along the lines of "do something nice with the pets instead of 'hurting the economy'" but some people aren't set up for that or don't have the incentive. If you want to see those pets given to someone and not deleted, trade that person and give them pets they may want or are missing. People are going to be selfish, there's no changing that, but if deleting pets is sooooo bad, you can try to curb the deletion of pets by giving people the incentive not to.
I am in full support of adding a button that's basically a "send to the pound" button, but I want to keep the option to delete my own pets should I choose to. They're just images and lines of code, they're not real, they don't have feelings...