That's incorrect, you
could change your A-ID to your new address at
https://appleid.apple.com if it wasn't already a me.com/icloud.com one.
...that's because me.com (along with the newer icloud.com) email addresses CANNOT be updated to a current non-me.com email A-ID – as they were, and continue to, already
be an A-ID automatically on their creation by the user.
(Which has been a pain in the butt for several thousands of Apple users like myself who were effectively forced to use two separate A-ID's for iTunes and iCloud, thanks to Apple not allowing non-email A-ID's to set-up and sign-into the iCloud services when they were released a couple of years ago. A completely ill-conceived thought process that one turned out to be, thanks Apple!)
So as you are already a me.com A-ID user, your only option is to add the gmail address as the alternate email, where you'll recieve any related emails from Apple concerning your account (eg. if it's used for iTunes, then receipts will go to the gmail), but you'll have to continue to sign-in with the me.com address, I'm afraid.
It's the same with everyone these days though... wanna use Google services: you'll need a Gmail email to do so. Wanna use MS services: you'll need a MS email to do so, AFAIR. The only thing I know is that at least Google allow you to sign-in on
some of their services with
any alternative email addresses they have for you on account which half-helps (although you can't choose which email address you receive notifications from, annoyingly!).
Sign-in's are becoming a big pain in the proverbial more and more these days, as companies fail to understand we don't NEED their damn email service, we're just trying to sign-in and do NOT want to get our comms from them to the email address they force us to have.
