I've had a number of big standards from that era with the same problem that, in my case, always turned out to be a case of badly rusted shift mechanism components. I also managed in every case through patience and perseverance (and a lot of Liquid Wrench) to free those mechanisms and restore them to proper functionality. 

Have you given the machine a good looking over yet? Are there obvious signs of rust anywhere?

Thanks for your helpful responses and ideas. Sometimes the best way to attack a problem is to set it aside for a while, and see what brainstorms might pop out of your subconscience.

That's what I did. Took a very close look at the shift-related diagrams in Ames & the HNSA version of TM37-305 (does anyone know where there is a downloadable pdf version of that document?), and examined the mechanism in another 1930s Underwood. It turns out that the problem was due to a slightly bent engagement flange underneath the left shift key mechanism that is mounted on the rotating rod connecting the shift keys. Because it was bent, it prevented rotation of that rod in response to downward pressure on the shift keys. After tracing the movement of the various parts that I could see (without disassembly) throughout the shift mechanism chain, it looked like that had to be at least part of the problem. A pair of broad nosed pliers, some gently-applied pressure, and voila! It now works like a charm!