OP, we rent out our property as we are overseas too and have done so for many years. Trust me when I tell you that it is a massively false economy to do so without doing so via a letting agent with a good reputation. Please don't even consider relying on a relative or friend - you need a professional who is well versed in the law and the vagaries of managing tenants and their issues.
Despite most of our tenants being fairly unproblematic - and many have stayed several years - it is still essential to have regular inspections with an agent on the spot who can resolve difficulties and queries plus also arrange for the required annual checks (Landlord's Gas Safety for example) and for repairs to be carried out. Our agents are fantastic and we completely trust them to manage our property effectively and professionally, and thank goodness we have had them on a number of occasions including various 'Pet Gates'.
To go back to your enquiry about pets. We always consider pets because we have them ourselves and rent overseas in properties where they are allowed.. We know how important they are to people and how difficult it is to find landlords that will permit them. Most of our pet-owning tenants have been fine, but we have been royally shafted on four separate occasions.
The first pet owners asked permission to have a cat (singular). We said yes. They got three cats. Who scratched doors, woodwork, carpets. And the tenants put a cat flap, without our permission, into our kitchen window which damaged the stonework and necessitated the complete replacement of the glazing bars and glass when they went. They fought over our bid to retain their deposit to repair the extensive damage and the DPS found in their favour despite us submitting numerous 'before' (cats) and 'after' (cat destruction frenzy) photographs.
The second had three very large, hairy dogs that they took for long, very muddy walks. Said dogs then were allowed to lie on the bottom of the long, lovely, expensive (because they are big bay windows) curtains in the living room without first being washed or de-muddied. When we took the house back the curtains were beyond cleaning or repair. They also placed the dogs' water bowls next to a Victorian wooden cupboard in the kitchen, up against the original Victorian deep skirting boards. Both were completely ruined by the splashing as they drank and had to be replaced and the room redecorated because the dogs appeared to have rubbed themselves against the walls when coated in mud. These tenants were lovely people but seemed completely oblivious and/or indifferent to the destruction the dogs caused. They did readily agree that some of the deposit should be retained for repair but the outcome was that we were over £2000 out of pocket due to the carpentry that was required, the redecoration and the replacement of the curtains. We managed to get second hand curtains that were suitable but they were still several hundreds of £.
The third lot bred sporting dogs - well we didn't know that they did and asked permission to have one dog. She was a bitch in pup and they then kept all the puppies. These lived in an old stable in the yard but destroyed the stable door by jumping up and scratching it, plus chewing the bottom and when they were permitted in the garden, created 'race tracks' where they chased round and round. You can still see the scars on the lawn and banks ten years later. And they also dumped all the dogs' poo - three years' worth - behind one of the hedges between our house and our neighbours' boundary. It took my DH and brother an entire day to remove it all - a disgusting job. And they too argued over the deposit but on this occasion the DPS found partially (to the tune of £200) in our favour.
Our most recent tenants are (or were) lovely but they got a 'lockdown' dog 'for the children'. As they both work permanently from home and did so before the pandemic, we said we were fine with that. However children and parents soon tired of the dog as it matured and they re-homed it, but not before it had destroyed the hall and living room carpets by chewing them and seemingly trying to dig it up... They have said they will replace it when they leave but as they have just stopped paying their rent on our giving them notice, because we now wish to move back in as my DH's contract is drawing to a close, I have absolutely no faith in their doing so and know that we are likely to face a long and expensive process to regain our property.
I know these are fairly extreme examples of the difficulties caused by pets - and we too have the 'pet clause' in our contracts - but you do need to be aware of what could potentially lie ahead. Do also bear in mind that some tenants won't necessarily be honest with you or your letting agent - for instance the multiplying cats and dogs I have mentioned. We have a dog and two cats in our current property overseas and none of them have created any damage at all, and if they did we would ensure to replace and make good as we know all too well how difficult it is as a landlord in these circumstances. We have also had tenants with pets who you wouldn't know had been there once the property has vacated. So as other pp's have said, it's the unpredictability that is the issue. They may have the most docile and non-destructive of pets, or one that does a fine impression of Road Runner on steroids..