Do you think it is a necessity to use a Bitcoin mixer to mix your bought Bitcoin before finally depositing them in your own wallet?
If you really like anonymity, then mixing Bitcoin from exchange to your wallet will increase the privacy of your receiving wallet. Remember centralized exchanges record the outgoing funds, and the receiving wallet will have a link to your account on the exchange.
make out transactions in between wallet to wallet or from exchange into your personal wallet or vice versa.
Or move bitcoin to an exchange, trade it to Monero, withdraw Monero and move Monero to another exchange to sell it to bitcoin. Withdraw it again. You don't need to use Mixers if you are ready to do some steps by yourself.
Isn't the identity of the address linked to verified account in centralized exchange traceable? Withdrawing to an address then sending it to another address isn't different from centralized exchange to own address except if mixer is use in between.