a stem recycling program is a good idea

To answer my own question: so far it seems I'm fine with a really short stem. I put on the itsy-bitsy teeny-weeny one (it SAYS 60 mm on it but is also sharply angled so it looks and acts even shorter) and tried it out on a long (75 miles) ride yesterday. It didn't seem to make handling too wobbly, and did make it easier to reach stuff. My neck still started aching after a while, but since I started looking for a new saddle I've become acutely aware of what's wrong with the one I've got, and spent a large part of the ride obsessing about that instead...

So I've got to get a new saddle before I start fussing around more with the bars, to make sure I know where I'm going to be sitting.