As someone who is allergic to cats the thought of having to suffer hives and hayfever like symptoms because someone is too cheap to pay to have it looked after, means I wont fly Virgin anytime soon.
It is the cats dander and saliva that causes the reaction, even in a carrier their dander can be deposited and we know they dont clean planes properly, and especially not between flights given tight turnaround.
Plus you just know some selfish owner will take the cat out of the carrier, and since they can sense when people hate them i know it would head for me.
Small dogs might be ok, provided the airports have somewhere for the dog to toilet before boarding and owners have an approved carrier that is lined with proper pad to absorb accidents. And if barks repeatedly on flight instant lifetime ban for the dog.
Owners also should have to pay extra (and the dog counts as the personal item), no carry-on + personal item + dog.
But how will they make sure the seat has room underneath, some seats have IFE box taking up part of the footwell. Only a flexible fabric carrier not a hard cage is going to fit.
I've only been on a plane once where pet dogs were on board it was flying BCN-SPU on Vueling. Dog + Carrier is supposed to weight less than 10kgs and must fit under the seat. I watched a guy destroy/remove the padding from the carrier cause his dog was on the over weight side, thankfully not sitting near me.
I have travelled a few times next to a Labrador guide dog SYD-MEL or MEL-SYD as when Id fly this route regularly there was often a blind gentlemen who was also a regular. Of course guide dogs are well trained and sit patiently at the feet of their owner. Never made a sound or had an accident.