Okay, it turns out that KDEnlive is heavily dependent on MLT, and in my computer's case it's MLT-6. I learned from a post on some other site to do an ldd on libmltavformat.so, which showed me that it was still looking for a pulseaudio library, all of which have been eradicated from my computer.
My initial response to that, of course, was to recompile MLT-6. That crashed with a nasty-looking error that felt like it would be way too much effort to correct, so I just went ahead and tried launching KDEnlive with apulse
Code:
# APULSE_PLAYBACK_DEVICE=plugequal apulse /usr/local/bin/kdenlive
and that worked. It's now using the recompiled binary that I created earlier, and it works just like it did before.
So, the moral of the story is, if you want to get KDEnlive to see libavformat, check your MLT installation. If you want to get KDEnlive running without PulseAudio on your machine, and you can't get MLT to compile, try apulse. It seems to work.