But if you value privacy, you should use DEX from the beginning, don't try CEX without KYC, sometimes there will be unpredictable problems, IMO.
I have been using CEXs without KYC for 7 years and have not experienced any problems. DEXs are not the best solution for everyone because they are very expensive in fees, do not support the right assets and networks, and have poor liquidity. I will switch to DEXs when one token to another is as cheap and hassle-free as on CEX, but until this happens, DEXs is not anything particularly unique that makes it worth abandoning CEX completely. 99% of the problems that arise due to working with CEXs are created by the users themselves. Whether this is the fault of the services themselves is a big question.
Being the early user it could've been possible. Right now it is really hard to make use of CEX without KYC. For the users who doesn't want to fulfill KYC, they could prefer DEX. With CEX things were going good and provides good liquidity. Majority of the problem happens with the careless of the user. Having a double check will keep everything functional without error. However while choosing the CEX we need to be careful finding the right one and using it, getting into the untrusted platforms gonna make the user suffer.