How to withdraw Gmail from iCloud account?

When I first created my Apple ID, I didn’t know anything about iCloud, so I entered my Gmail (nickname@gmail.com) as my primary mail.

After this creation Apple gave me iCloud mail (nickname@icloud.com/nickname@me.com) which now I want to use as my primary mail and forget about Gmail forever.

Problem is: I can’t remove Gmail from my Apple ID and use iCloud mail instead, using appleid.apple.com. I’m getting message that I couldn’t "use a domain owned by Apple".

My goal is to withdraw Gmail from my Apple ecosystem and use in it only iCloud mail as primary mail for everything: for login, for getting messages and notifications from Apple, everything.

Who can help me with that?

P. S. The main reason is that I want receive important emails from Apple on my iCloud mail, not my Gmail, because I barely check my Gmail and almost forgot about its existence.

So if there is a way to make Apple send notifications not for primary email but for alternate—it would kinda solve my problem.