The problem is bad tenants can rack up £1000s in damages very quickly. Insurance companies will often not cover, also security deposits are usually no more than two months rent.
I had to rent out my first main home a few years ago for 12 months. A nice two bed starter home. It was let out to two young women, who had good references. They asked if they could bring their one, older dog. I said yes (I love dogs, and DH and I now have our own spaniel).
The property was let unfurnished. It was agreed they'd pay for a deep clean (if required) and any damage (if it occured).
Great right? (House was let for £575 a month, deposit was £750).
Except, without notifying me, they bought a second dog - a puppy. This puppy was left alone for 10 hours a day plus, it was not trained. It peed everywhere - which was never cleaned (ruining all carpets and underlay) it chewed everything - doors, and even right through the bullnose on the stairs.
The house needed not only a deep clean (which they never bothered with) but also recarpeting, rewallpapering (because boy puppy had peed up the walls) and new doors.
That's before we go into the poo clean-up in the garden.
It cost well over £1200 to out right, and that was with my elderly dad and me doing the work and finding cheap end of rolls for the carpet (thank heavens my dad used to word for Allied Carpets of old). The damage would have cost more than £3k to put right if we had not been able to do that.
That's why landlords often say 'no pets'. Because we allowed a pet, my landlord insurance would not cover the damage (even though I had stipulated exactly what I was doing).
There are many, many awesome pet owners. We had our dog as a puppy in that house before we moved last year. She did not pee on carpets (the odd accident in the kitchen and lounge though - but this was cleaned up immediately!) But then we didn't leave her for most of a day on her own.
So many places do not allow pets, so it's very hard to get a pet reference from previous landlords. And while an increase in security deposit sounds ace £3k for a £575 a month rental is not going to be achievable. Let alone what it would cost of the place had been furnished!