Furnace keeps trying to ignite but never does -- no codes
Thermostats sometimes do fail...

You know how we keep saying that HVAC problems usually aren't the thermostat's fault? Well, it turns out, whatever hit the original control over the head took out the 'stat (a Honeywell CT3500A4453 from 2007) as well, but in a subtle way.  It appeared to work -- it responded normally to button pushing, and was trying to command heat.  However, the W wire output from the 'stat was sagging (down to 21V from the normal 24-28V), and this was the cause of the sagging voltage going out to the gas valve, as jumpering R to W via my ammeter at the 'stat end of it all caused the furnace to fire up without a problem.  Replaced the thermostat with a cheap non-programmable one, and problem solved!

Further analysis of the situation showed that the control board, instead of generating its own 24VAC signal to the gas valve, was switching the 24VAC W signal from the thermostat, which explains why the thermostat's sagging W output was causing the gas valve to malfunction.  Furthermore, while I could not prove it, the only likely candidate for why the thermostat was misbehaving would be a damaged relay contact on the thermostat's control relay.