We have a "York Diamond 90" gas-fired furnace that was working well
until today.

There seems to be something wrong with the gas flow/valve, burner, or
sensor.

I can see the electronic ignition element turning white hot, then hear
the gas valve click on, then see the blue gas flames light for a
second or two before sputtering out. This cycle repeats itself several
times until, the system "gives up" (or I guess locks out) at which
point the blower just continues to blow cold air forever and no
further ignition attempts are made. The blower continues to blow even
if I shut off the thermostat

Interestingly, if I cycle off the power (even just for a second or
two), the furnace will start up normally and ignite the burners
properly. The burner will continue to fire for a *couple of minutes*
before blowing out at which point it goes into the (failed) cycle of
trying to relight a couple of times before finally giving up and again
leaving me in the stuck state with the blower blowing cold air.

Sometimes in the cycle of trying to re-light it will burn for a
few seconds before sputtering out. Also, sometimes, I hear the valve
clicking on-and-off a couple of times in rapid succession.

Now the other two gas furnaces are working properly, so I don't think
it is a supply problem. Also, I don't think it is a ventillation problem
because it seems to burn fine for a few minutes with a nice blue flame
before suddenly sputtering out.

When I opened up the bottem panel (overriding the disable switch), I
noticed that after it locked up in the continuous blower position, the
diagnostic LED exhibited a pattern of 8 short (red) flashes -- I believe
this simply indicates that the flame is lost 5 times within a heating
cycle which according to the online manual indicates either:
Low gas pressure (unlikely since other two gas
furnaces in the house work fine, right?)
Faulty gas valve
Dirty or faulty flame sensor
Faulty hot surface igniter (unlikely because the flame does
ignite, right?)
Burner problem

Any idea what might be wrong and how to fix?

Even if this is not a DIY fix, I would like to have some idea of what
the potential problems could be so that I can intelligently engage the
service person and make sure that I am not being taken

Thanks!