The plan was to be farther along with this restoration, but I think I heard a saying about the best laid plans...
I came away from this exploratory surgery with a fairly good idea of what was causing the wayward keytops. Unfortunately, the solution wasn't the button I found deep in the bowels of the typewriter. That would have been too easy.
The keytop arms of this particular typewriter are very long. The extend all the way into the back of the frame where they pivot. I think this is something well-known to Underwoods. Each bar has a small tab of metal that engages with a comb/pivot plane. An extended "Z" bar-like piece covers these tabs and the pivot comb allowing the arms to stay in place, but also move freely. Small springs add some push to the tops and keep everything aligned. In the picture below, I have removed the retaining bar for ease of viewing.
On this typer, several of the arms had worked their way out of the pivot comb. Their conspicuous absence can be noted above. This small change rendered the typewriter unusable. Here is a better view of the absent key bars: