I think if someone uses a proxy to access these sites when a site doesn't allow them to access then site may ask for your documents. But I too agree with that, if they detect something like this then they shouldn't allow for deposits until they provide the required proofs instead of blocking their withdrawals.
What's wrong with using proxies ? I wouldn't like to expose my location details and I'm sure free to access the website however I want.KYC should be used before a deposit is made and not after an withdrawal is requested.
I'm not sure whether it is easy for site owners to detects these deposits because they may check manually huge withdrawals but it looks all deposits are automated.
So they should be tracking deposits as well.I don't think it is very difficult to do so.