I've got to agree with the animal rights groups. Maybe not for the same reasons though.
An animal can be damaged by an irresponsible renter- killed, traumatized, injured, de-socialized. Unlike a rental car or TV, we cannot simply throw out the broken ones (I should hope you agree with me there)
Complicating things even more, the odds that you're not going to know how to interact with a pet go way up if you don't HAVE a pet!
While animals are not people, they are similar enough to people as to be completely different from 'stuff', and if you cannot deal with the responsibility of caring for a pet I do not think you should have one at all.
(In a controlled environment that can include supervision and a responsible figure who knows what they are doing, perhaps it could work, but not if you're taking them home for a week or such)