OP is there a "keep current" affordance which updates constantly to show real-time positions?

I assumed the "play" button would do this; what I see is it pulsing with 1s/s and || indicated which I guessed meant "this is now PAUSE; and, update is one fps"...

...but even selecting e.g. a Starlink sat I do not see any motion.

Hug of death, or, misunderstanding of feature?

Not a hug of death. It runs live at 1 s/s, but at real time a satellite moves only a pixel or so a second. The middle button toggles pause, so your click paused it. Press it again, or + to speed up.