What that is most important is to avoid giving your identity documents to people. Those sites too can misuse your identity documents that you submitted to them.
But if you do not give identity documents in these big websites or exchanges, they will put limit or restrictions on your account. Even to do P2P transactions on Binance, some sellers and buyers demand identity documents and they will deny the deal without these documents.
I think you got him wrong. To use the Binance p2p platform you don't have a choice, you must verify your identity before you can start using it. So, this is not a choice. You either comply with it or find another p2p platform.
On binance p2p, merchants don't have the right to ask you for your personal documents since Binance has already verified them. Any merchant who asks for such docs or any other personal information is mist likely a scammer and better avoid dealing with him. If he refuses to complete the deal then simply report him and binance's customer service will take care of the rest.