How to resolve Apple Home account issues with iCloud settings?

I'm trying to fix my wife's iCloud account specifically around the Apple Home app. Here's the situation...

In Apple Home if I go to Home Settings it shows that she is logged in as her @icloud.com account, but that email address is not the one she uses to sign into iCloud, it should be her @gmail.com account. In fact, she never uses the @icloud.com email and it was mistakenly created some time ago.

When I look at the iPhone settings it shows her Apple Account as the correct @gmail.com one.

If I go to the account.apple.com site I can see next to the @icloud.com email a ? mark that, if I hover, says, "This is an alias of your Apple Account primary email address." And it shows that the @gmail.com email is the primary email.

So, my immediate problem is that I want to change the account that is being used for Apple Home, but I don't see any way to logout are choose which account is used.

Additionally, I would like to get rid of the @icloud.com email address entirely, since it is not used and is just causing confusion.

Can anyone help on these?

[Re-Titled by Moderator]