Turns out there wasn't an obscure blockage or broken motor preventing the furnace from behaving. A capacitor was failing, so the blower motor couldn't start, and that kept the furnace from completing the start-up process.

The furnace would start up the fire within (which did actually fail before, so there was a blockage), but then fail to kick in the large blower motor. I found that (dangerous, I know) if I gave the blower a little push on one of the vanes while it was trying to start that it would fire up just fine and all would be well.

I finally broke down and called a repair guy who finally came out today to check it out. He all but diagnosed it over the phone; if I had any idea where to purchase the capacitor, I would have tried that without his aid. He showed up, gave it a quick diagnostic, verified my fix worked, tested the capacitor with a probe, and a couple minutes later all was well again.

While he was there, he noticed that the electronic filter was installed backwards. I commented that the installer had noted that, but also said it wouldn't be a big deal. This guy told me it was a big deal, and with a few flicks of the screwdriver had it corrected. Seems the unit itself is bi-directional, and that the little block on the bottom (to prevent installing the filters backwards) could be unscrewed and put in the other hole. He made a quick switch of the pull-out handles on the filters and slid them in the right way. All included.

He also noted that the plastic pre-filter I was using was probably dangerous. I noted that I'd looked for replacement metal filters, like we had originally (but that had been damaged beyond use), but they apparently aren't sold in stores. He commiserated and said that he'd order some for me from the shop (and bill me later).

The part cost $15, the visit $95. In all, it took about 45 minutes, but since I no longer have to race down stairs to help kick in the blower motor (and we won't wake up to 50F temperatures any more), it's worth it.