Delayed action stabilization!
The Stabilization section in the Inspector was greyed out in spite of the tick box and the clip in the timeline being selected.
I fiddled around a bit to no avail and finally found an old thread from this forum where someone had a similar problem but said that if he used some other app and then went back to FCP the Stabilization was working.
At that point I closed Safari and sure enough, the Stabilizer was working.
So I tried it with another clip and that was greyed out, so I just sat around thinking about it for a couple of minutes when suddenly out of the corner of my eye I saw the image in the Viewer "jump."
The Stabilizer in the Inspector was no longer greyed out and everything was working as normal.
So it appears that on my system (and also the OP's) there is a delay of a few minutes before the Stabilizer starts working.
Any ideas about this strange behaviour?
iMac 27″ 5K, macOS 13.6