In no way coinbase or other centralized exchange will help you to remain anonymous as almost all of them will have enough information that may lead up someone knocking at your door also if you think about nunber of data breaches in those platforms risk of getting your privacy blown out is very high.

Possible solution include use of multiple instant exchange platform that doesnt ask for verfication and sending those exchanged alt (eth) to multiple addresses (created as paper wallet) than using dex platforms to gain wrapped btc than exchanging it for real btc before sending it to one address. Also never use same computer/browser/ip/network in all of this exchanges.