But I'm not sure, if the senders address would be visible on Etherscan or not.
The Blockchain link may seem to be broken but there will still be a link of internal transfers between your Binance accounts. So that is no privacy at all
You could alternatively use p2p exchanges that don't need any single KYC document. Bisq is one of them
Many more exchanges listed in this site -
https://kycnot.me