You either give up on having an anonymous email because they WILL FORCE YOUR real name to be attached to your email, or you give up on Google integration features because you simply CANNOT use Google features without a real name anymore.
Trying to switch between my "public Google account" for functionality and my "private Google account" for privacy is almost impossible when you consider how broadly Google integrates the features. It's simply unteneable, completely. Google wants my privacy gone, and is denying me access to their services until I agree to give it up entirely.
Very, very frustrating for users who have more than one account.