Thanks, that's it. I really like 'gradually walk the UI tree', though I'm struggling a bit with the thing I want being obscured by the label (and me not remembering what was there). An offset would be nice, though I'm sure it's not easy to define a heuristic that works in all places.
The first thing I tried was in my browser, I asked for all the buttons and it labelled the 'x' to close each tab, but where the labels were laid out it just looked like an inviting right-aligned label on the clickable tabs themselves. Lost a few tabs before I realised, because you also don't see any feedback on the click since any UI feedback is hidden by the label. Hmm. Food for thought.