Oh fuck. I love Hide My Email and it's been the best feature about iCloud ever since it came out.
It's actually useful compared to Gmail's useless "yourrealaddress+alais" that gives away your actual email anyway, and it helped me catch quite a few spammers/data sellers.
Hide My Email addresses already have a peculiar format that others could guess, and some do block those, and there's no reason to add a blatant "private." tag.
This is a win for privacy-intruders, not users, just like Apple's iCloud Keychain API that has allowed Facebook, TikTok etc. to secretly track users across multiple devices and device reinstalls for years.
FWIW it's not a gmail thing for privacy, but rather just part of the email spec. RFC 5233 talks about it.
https://www.rfc-editor.org/info/rfc5233/