You should remove Gmail as as service from your Google Account in order to be able to use an email address from a non-Google service as the Google Account primary email address.
Can the primary email of a Google account be changed to something other than the Gmail address?
How to set my alternate (non-Gmail) address as the account primary?
Gmail (for email addresses hosted on gmail.com) allows "plus" aliases that doesn't require a verification email. For all other "aliases", specially those added by using "Settings, Accounts and Import, Add Another email address" requires a email verification. Why? That is a Google policy.
You can check out https://tellspin.app.
It's a Slack app that automatically rotates users within a Slack user group (or alias) based on a schedule. You can have an @oncall user group updated daily or weekly with different users depending on how you configure it. It can set overrides too.