I struck gold (as respects my issue, anyhow)!
I found that I had lost HTML5 capability not just on YouTube, but everywhere, and that appeared to have happened simultaneously with my updating to the latest version of Flash, so my first stab was to download the Flash uninstaller, and that resulted in my getting the dreaded "Your browser does not currently recognize any of the video formats available." screen, which, in light of
this , made no sense at all.
I then realized that although my symptoms differ from Rita's, they are similar, and we're both running "ClickTo..." Safari extensions...she, ClickTo Flash, and I, ClickToPlugin, so I disabled mine to see what would happen, and VOILA, I regained HTML5 capability.
I then reinstalled Flash (I don't love it; I need it to list on eBay.), and HTML5 still works.
So, my suggestion to Rita is that she disable ClickToFlash and see what happens.
More: It would seem that I no longer need ClickToPlugin, because HTML5 is my default player without it.