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> Do I first have to somehow connect it with my Google account (how?) and then Groups will know about it

That turned out to be the key.  After a lot of poking around, I went to "Manage your Google Account" (from the user avatar on any Google site), then "Personal Information", then "Contact info", then the "email" section, and inside there is a section called "Alternative emails" ending with a link "Manage alternate emails".  In *there*, after another password check, you can click on a button to add another email address to your account.

I then had to wait a couple days (I don't know if that's normal for Google), and *then* I could successfully follow the instructions in the documentation to change a single group to use that new address.  You have to do it for each group (no global option).  I've changed it for a couple lower-priority groups and now I'm waiting for messages (to confirm that they went to the right place instead of disappearing into a black hole).  When I see those groups work with the updated address, I'll edit the important ones.