As described above, I created the 2nd connection, and am the Owner.
Also, the connection is available when choosing the connection to use for a template. As described above, I was able to use that interface to change the connection for the template, after which it became the only connection available to use in the Flow.
Unfortunately, even though that would seem to indicate progress, when I test this Flow it fails due to Unauthorized.
So now my state is that I can select the connection I could not previously see, but despite that the credentials don't seem to have travelled.
This just seems very broken.