I think all the things you mentioned like deposit and ID are mainly about security, both financial and physical safety. There are a lot of time wasters with privates so a deposit shows intent to go through with the booking. I think that booking a private is a luxury. You're not just paying for the time of the booking but some of the exclusivity. They are inviting someone who they haven't met to see them, just as you are wanting to find out if you want to see them, they want to find out if they want to see you. I'd be screening if i was a private
Here is some of the appeal I have considered:
I don't have to hope they will be available.
The incall location may be a nicer setting than a shop.
Its a better mood/vibe than a shop.
There is less chance of awkward intro room interactions with fellow punters or the chance of being spotted entering/leaving the shop.
Having social media accounts means that I can get a little bit more of an insight to them.
I can text them beforehand to gauge what their service is.
They may see less clients.
I feel like this thread could turn into a discussion where someone says privates are scams, or they are not worth it etc. In my mind these topics are similar to getting a steak for dinner. I could do it myself, I could go to the pub or I could go to a fancy restaurant. All are steak, some might be better than others but at the end of the day its my choice where I eat so its subjective to the client.