The two exchanges that we would recommend would be Kucoin and Bybit.
We wrote reviews on both these exchanges for anyone interested:
https://www.thecryptobreakdown.com/exchanges/kucoin-reviewhttps://www.thecryptobreakdown.com/exchanges/bybit-reviewBut the TLDR is that both of these exchanges allow users to deposit and trade cryptocurrencies without KYC.
And they both also offers a wide range of cryptocurrencies, 700 for Kucoin and over 300 for Bybit.
Having said that, we would recommend that you do your trading there and then withdraw your funds to a non-custodial wallet.
I can attest with Kucoin as I did trading on this site several times and didn't encounter any withdrawal problem.
But yes, your suggestion of withdrawing the funds after you trade is a must in every exchange.
Because no matter what the reputation of the site is, you should always keep in mind that it is always best to secure your coins in your own wallet.
I think there are still several non-kyc exchanges, however, topping my list is also kucoin. Bybit, haven't tried this one yet.