Picking any exchange would be hard, I will give you that much. Even if you are dealing with places like Binance or Coinbase, it could become a difficulty because they require KYC and they could still get hacked in a way that your KYC could be shared with the world.
KYC should be serious problem for us, and why we have to worry about this policy ? isn't we want to maximize our security layer to protecting our account. and passing KYC could help to protect, maybe hackers will have trouble when withdrawing our assets but have different name on bank account and receiver. honestly i am feel comfort if asked to pass this stagge, beside for security reason my account limitation could increase.
There is a double edge sword in that situation. If you give your KYC and then report to the exchange that your account is hacked then they could stop the account and then ask you your KYC and you give it to them and they will open it to you only and you will get your account back that way.
This is the great part, which happened to me (sort of) where I forgot my password and didn't had access to my email, so they asked my KYC and I sent them a new picture and they accepted it and I got access. But the downside is the one that I talked about, it is risky because we may not be the ones that get hacked, it could be them that gets hacked as well, which is a problem.