I personally do not use CEX.IO and have long since left it, occasionally using coinbase and indeed at that time they asked for a copy of ID and I agreed to it but after that I could be comfortable using coinbase, but it was not too often transacted there for fear of some negative feedback. But, if you are a person who is reluctant to send KYC, then I guess that will make you worry, because some large exchanges require us to give KYC and so far I don't think that is a problem.
KYC is something that we really cannot shy away from, because it would be worse than this in the future when there is full regulation on cryptocurrency generally by the government. kyc was never a priority of these exchanges before, but I guess they have seen the future and they know already that it would cause them big issue when cryptocurrency becomes regulated and it would be enforced by government.
The result of the importance of KYC to government is what has led to the things we are seeing now that virtually everyone is requesting for KYC now. So the op just need to get used to dealing on exchanges with KYC, there are reliable exchanges that would never do anything bad with our data and I can beat my chest for exchanges like binance and some other ones.