I keep returning to, the release of pets per page needs to be drawn out far longer. Maybe even add pauses in between pages or something. Maybe for the pauses, have something like "We need time to gather more pets, please come pack in x minutes to come again) and then release another few rares after that.
In my opinion, the pound, even on events, should never be closed in the first 2 to 5 minutes of opening. It doesn't need to be open for hours, but the window of time to adopt a pet I feel should be a full 15 to 30 minutes to get something rare to extremely uncommon and 40 to 50 minutes to adopt common pets. By drawing out the process, this slows down the rush within the first 50 seconds of opening and forces people to wait longer. It is also a hope that this will spare some server load and prevent a lag or at least even traffic out.
This time was so extreme that I clicked, waited, there was no sign of the page refreshing. I had to hit the menu on mobile to go back to the pound to reload the page, just to see 90% of the pets were already taken. Then try again, have that repeat.
On the 4th refresh had 140 pets left, clicked a common and then got "There is a problem with the pound, please go back and try again" then found again a page full of taken pets. Refreshed again and there were only 14 left. Refreshed again and all but two were taken.
(I have a large computer screen, so I bolded text of the points I feel more strongly about so they don't get lost in the rest of the paragraph)