UPDATE:

The heater has been working (although it has been almost 70 degrees
lately) just fine with the new flame sensor.

Apparently, this is what happened:

1. Heater wasn't working for so long that it "forgot" the prior
error code, so it was just blinking incessantly.

2. Eventually, the heater got around to blinking a code for a
bad flame sensor circuit.

3. The observation was the flames would light and then shut off
within seconds, four times, and then lockout would occur until
the power to the furnace was turned off and back on.

4. All measurements to determine whether the flame sensor was
broken had failed, mainly because 5uA is hard to measure and
because the meter I had borrowed must have a blown fuse.

5. However, replacing the flame sensor worked.

How the flame sensor failed is a mystery, as it clearly was not
tarnished (it was sanded with steel wool, emery cloth, and a file,
over subsequent attempts). The ceramic "looked" intact, but may
have developed a fault such as a crack.

The assumption is that the flame sensor failed, although the
only proof of that is that the new flame sensor worked as
soon as it was installed.

Thanks for all your help and advice.
Who knew that the flames form part of an electrical circuit!