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.
As I know, most CEXs require KYC before we can use them. Only a few do not require KYC but some features will be limited and withdrawal amounts will also be limited. I am not saying that DEX is superior to CEX, DEX does not have full functionality and has many shortcomings such as fees or liquidity that you mentioned. That is why to date investors and day traders still choose CEX over DEX despite collapses like Mt.Gox or FTX. But if someone is just a pure bitcoin holder and cares about privacy then DEX is the perfect choice for them.