I had an excellent MSFT support technician help me out with this problem. The advice we've all heard - that you, as a Global Admin, must create an external user and make them a Global Admin - is not correct. You must instead, as a "Global Admin", create an internal user and make them a Global Admin. That Global Admin user should have the access that you, as the sole owner of your own Azure subscription, and supposed "Global Admin", do not have. Note my very selective use of quotes.
I asked the technician why this Byzantine circus was necessary, and he explained that, for most purposes, the owner is a "User" and not a Guest. However, in some dark, weird, way, this is not entirely correct. Owners are actually Guests with most "User" privileges. Yes, if you look up your user account, it plainly says "User". And I'm telling you that's not entirely true.
So now the workaround makes sense. We make ourselves "Global Admin", create a user as a real, bona fide Global Admin, and use them to log into these websites. Call it "security by obscurity" if you're feeling cynical.