In the process of binging other methods, I found Freemake a (free) video converter which just hit 3.0, and added the ability to create HTML5 video. And HTML5 video with burned in subtitles that almost works!
(I can view the video on DroidX, iPhone, Chrome, and IE. For some reason Firefox is having trouble. VLC can read the ogv file.)
Summary: It did what I needed. Soft subtitles* would be nicer – but the fact that it handles them at all is cool.
* “Soft” subtitles, or soft subs, are subtitles applied at playback time from a subtitle datafile, either muxed directly into the video file (.mkv, .ogm, etc.), or in a separate file (.ssa, .srt, etc.).
(citation from http://en.wikipedia.org/wiki/Fansub)