Called Apple support. I'm not sure I understand the "why's" about this, but they said to start up the Mini with cables connected - which was what I did at first, but since then I had been switching cables, ports, and TV without doing that. At this point the Mini had shut itself off but cables were connected to old TV. Turned it on and it showed up. So moved the cable to the LG TV and got no connection. Forced the Mini to power down and then back on and it worked.
But the fun wasn't done. They wanted me to go through the initial setup process which brought up a rather big problem - you have to type in the passcode for your wi-fi. Oops, I didn't have a wired keyboard and no way to pair before setup! Seems like a weak point in design. If they would bring up an onscreen keyboard, I could have gotten through it.
Then I remembered the TB and dongle. I actually have two, one for each TB I have. Didn't remember which was which so the first one I tried didn't work. The other did, but right now, with all this stuff on my desk, the Mini is sitting back behind other stuff so, usually, to type I had to walk to the other side of the desk, type (not being able to see the screen), then come back and hope any passwords, etc, didn't have any typos - especially those which don't actually show a passcode (just the dots!). Very frustrating.
Cable TV still doesn't work. Maybe it's coincidence and that system is just down for everyone. Have to look into that.
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