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This is pretty standard. Fan wires generally are only used when you want to run the blower without heat or ac. When heat is called for, the logic on the control board handles blower operation, as the precise order/timing will vary from furnace to furnace. On mine, for instance, when the thermostat calls for heat:

The small draw/exhaust blower runs for ~30sec to clear the combustion chamber

The ignitor heats

The gas valve opens and combustion starts

Wait~20-30 seconds

Then the main blower starts.

If your furnace is short cycling (shutting off before reaching the demand temp, despite the thermostat still calling for heat) this issue is likely at the furnace. Have you replaced your filter recently? Too low of airflow over the heat exchanger can cause a safety shutdown. Another common cause of short cycling is a flame sensor getting gunked up.

HVAC service tech here. If you are interested in having your fan run continuously (which I recommend) it can be done using your set up provided that the new thermostat can run on batteries. If it can use batteries then the green wire that goes to C can be moved to G in the stat as well as in the furnace cabinet.

Maybe it's just me, but starting at the thermostat has always been baffling.

I always start at the furnace, figure out what's connected to what down there. Informed with this I can head upstairs and decode the thermostat.

Eh, I started at the thermostat because I'm a systems integrator and the first problem solving step is "Turn it off and on again". When I flipped the 'heat/off/cold' selector lever on the thermostat to 'off', the LCD display declined to switch off the 'heat on' section. If you touch the non-touchscreen display it spazzes out, and so does attempting to access the user or installer menus.

Lastly, the furnace would work correctly if the thermostat was popped off the wall, batteries popped out, then replaced on the wall until it reached the demand temp. Once it got to the demand temp first time, it would stop working again until the thermostat got power cycled or at least removed from the wall again.

But really I just tried to turn it off and on again and it wouldn't turn off properly.

And I may have misread your comment. If you're talking about establishing the wiring patterns, I can sort of see that, but on the other hand it's kind of like going back to the patch panel to see if it's wired for 568A or B instead of just looking at the socket you're switching out. Feels like a wasted effort if everything in the wall box makes sense.

Have you replaced the batteries?

Thank you very much for the response, it was informative and helpful. We were actually both right - my thermostat is having problems and needs replacement, but they're all user-interface and display panel issues. Getting the heat back on was just replacing the filter (Which I realized when I went to check it today was last done 18 months ago. Where does the time go?).

Yep. Couple days back when it started really doing it enough to convince me that the furnace wasn't just struggling to keep up with the extremely cold weather.

Yep, a couple different approaches.

I'm usually brought in at the 'I tried to install this nest, and it isn't working' stage of troubleshooting, so it's already been fiddled with and can no longer be trusted.