It wouldn’t be the shelter picking up a dead cat, more likely Animal Control. They would take it back to the shelter, scan it (no chip), take a photo for ID purposes, then send it to the local pet crematorium. No chip generally means the owner doesn’t really care what happens when the animal goes missing.
There’s no need to go in if the animal is clearly not yours. In this case a question as simple as “does the cat have white paws?”. If not then it’s clearly not Rarity. Shelters don’t need random people wasting their time on an unnecessary in person ID. As
@brooky said, they aren’t hanging on to dead animal bodies on the off chance an owner might come looking. They’re headed straight to a mass cremation.
Edit: realised I’d linked the wrong user.