> - Label Hide My Email with the service name I registered with it.

I think the ‘normal’ way to do it is way simpler:

- a site asks for an email address

- click “Hide My Email”

- use Apple’s flow to create a new email address

- use Apple’s flow to pick a password

- phone or Mac automatically associates the email address with the site and stores the password in the KeyChain

AFAICT, the only thing that doesn’t do that you describe is “Add an email rules to move the email addressed to that Hide My Email address to a separate inbox”.

I think that’s orthogonal to using Hide My Email, though. If you want that, you likely would do the same for mail from the site’s domain.