I'm not really sure how Apple's Hide My Email works, but my impression is they work by creating a proxy email for you. If that is the case, it should be a good solution for protecting you privacy. The problem is you become hostage to Apple, because now if you loose access to your Apple account you also loose access to ALL your accounts(potentially). It's probably on the same level as using a password manager like BitWarden.

I've just explained the problem with the gmail tagging in another comment.