It happened again, so maybe I can provide another data point or two. If I configure the display to be "only on 2" (the larger), then the search tool is on screen. I can then go back to "extend" and see the search tool.
In the new config, the search tool was on screen. So it was not duplicating today.
In the bad case, the "find and replace" tool was checked and I could not get it to uncheck. Selecting there tops it but that doesn't help when it's off screen.
This never happened on Win10, so I'm guessing something about Win11 lets it happen. But it doesn't always happen. It may also have something to do with hibernation. I hibernate using PsShutdown64 -h, from the Sysinternals package. Maybe it moves off screen when I hibernate with it open? I haven't confirmed that.
Anyway, there's a work around, so I'm fine and I hope this report can prove helpful at some point.