I don't seem to be able to find a "Check edition" button or command. However, the "About Publisher" command (on the new Win10 system) gives the following:

As you probably will surmise, I was able to use the original installer program to get Publisher 2019 running on my Win10 installation. There was an option to enter the product key, which I did. Publisher started and I was able to open and edit a .pub file.
Next time I started the program, I got the activation wizard. Using the default "activate over the Internet" I get an error that the activation key is invalid. No other details.
I'm unable to see what the program thinks is my activation key, so I cannot compare it with the physical paper activation key. But I'm pretty sure I entered it correctly.
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Possible news flash: The issue may be that I've downloaded a 64-bit version vice 32-bt. I just looked at the "old" Publisher, and the "About" info is different:

I'm willing to redo the entire Win10 installation if there's a different installer that might work.
(And again, everyone, many thanks for your shared wisdom.)