@Mike-And-Jane0
What I have been told by several Airbnb reps is that third party bookings are discouraged on Airbnb but that it's up to the host to accept them or not. However, when you remind them that the booking will no longer be covered by Airbnb should anything go wrong, they confirm that, that is correct.
So, it's no skin off Airbnb's nose. They still get the booking fees and they don't have to dish out anything for damages nor get involved if guests break rules etc. I am not saying that they are necessarily always going to help if it's a regular booking, but I'd be willing to bet you won't get any help at all if it's a third party booking you've accepted. If you're willing to take the risk, then that's fine. I know not all third party bookings are problematic/dodgy.
My issue with it is that your agreement is with the booking guest. I've never hosted a third party booking to my knowledge as I always tell them the person staying must book. However, I have hosted people (parents, spouses etc.) who had someone else handle the booking for them using their own account, so I didn't realise I was communicating with that person's husband or son or whatever.
EVERY time, that has turned out to be a problem in some way because the guests who arrive often haven't read the listing, they certainly haven't read the house rules, let alone agreed to them, they often have unrealistic expectations and therefore can leave dissatisfied and they frequently break house rules. No, actually, they always break house rules, big or small, because they haven't read them!
I had one guest who let her husband book all her stays via her account. He pretended to be her, agreed to the house rules etc., but it turns out just showed her the pretty pictures and she said, "fine". She had not read anything, immediately started breaking rules, stated bluntly that she was not even going to try to follow others because that's not how she did things at home and complained the heating wasn't on even though it was summer.
Another guest booked for his mother, using her account again pretending to be her and, even though the booking was just for his mum, made full use of my kitchen, garden, bathrooms etc. during her stay. He seemed to think that because he was the one who had booked for her, he was also entitled to use the listing. It was a short stay and he didn't sleep here overnight, so I let it go. Most of these circumstances were where the people staying couldn't speak English. That also means they didn't bother to review.
Another problem with third party bookings is that you are looking at the reviews and verifications of the booking guest, so you really have no way to judge what the actual guests staying are going to be like. The person who leaves the review hasn't even stayed, so chances are that they aren't going to bother to leave one if all went well or write some sort of inaccurate nonsense if it didn't.