I think something like this would be a fantastic idea. I do think there's way too many low color pets floating around that folks (myself included) just don't want.
The forest really doesn't have enough of an incentive to toss pets into it, so maybe they could update it with your suggestion of non-monetary to item based rewards.
Edit: meant to add to the forst thing. Now that I think about it, a forest update would be a double-edged sword. Users could still opt to keep pets in the game contributing to the too many pets thing but at the same time, with 10 turns each and a better reward, there might end up being too many pets for folks to try and keep them all in circulation. Maybe if they do update it, reduce turns?
Idk I don't wanna hijack your suggestion.