I think the last few pages of discussion have shown my issue. I have an LD monitor with HDMI 1.2, not 2.0. It is incompatible with startup for the Mac Mini 8,1 and probably always will be. (It's a $340 monitor so I'm not happy about possibly junking it.)
Apple does say right up front that they only support Thunderbolt 3 or HDMI 2.0. Well, they don't really say that they won't take HDMI 1.2 or Thunderbolt-to-DisplayPort, but they also don't say they will!
They do say that their own TB3 to DVI should work, but the monitor only takes DisplayPort and HDMI 1.2, so that doesn't help.
I can, if I rearrange cables, start up with my old HD Spectre, running from the HDMI port into a DVI adapter (I don't think it works if I just connect via HDMI but will try that next). That works, then I connect the LG 34UD HDMI 1.2 to a TB3-to-HDMI adapter.
Realizing that HDMI 1.2 is not supported (and probably never will be) is essentially the end of this, I hope. I still sometimes have the no-monitor-on-return-from-sleep issue. Going back to sleep and waking up again works to solve this.
Not really plug and play or “it just works,” but Apple hasn’t promised either of those in quite a few years. I guess it looks good and it's thin, and that's what really matters, right? I do like the speed, and waiting for the new Mac Pro wouldn't really have gotten me anything.