Are you trying to remove the label of your personal info from your wallet? Hmmm I don't think so? Unless you start from scratch. Granted exchanges would have info about your past wallet, but they'd have no info about your current one as long as you don't do KYC again. You also won't transfer funds from your old wallet to you new wallet since that may lead to them tracing you and they can safely assume that you did something. Checking your next transactions then and one could easily assume whether the new wallet of yours was a merchant or really yours.

Still tbh, there's no problem with buying BTC from exchanges if you aren't that paranoid about privacy. You've been basically living you're entire life like that anyway.