The HVAC technician came by, though could not replicate the error condition when it was 44F (6.7C) outside, perhaps because the error occurred in sub-freezing conditions. Additionally, I found out the interface of my Goodman HVAC system motherboard has a proprietary communicating protocol, using R, C, 1, and 2 (not L1 and L2). The HVAC motherboard also has sockets for G, W1, W2, Y1, Y2, and O (see below), though they would require a different plug, and I have no idea if the firmware on the chip currently supports those connections. The HVAC system has a 2 stage heat pump and a 2 (or 3?) stage gas furnace.
HVAC system motherboard interface
Thermostat mounting plate
So I contacted Ecobee technical support, and after significant back and forth of photos/etc, their analysis concluded that I would have to call out a competent HVAC technician who could set up the correct plug wiring, and snake the new wire bundle through the walls to a new Ecobee thermostat, and there would be a good chance it would work.
So with that estimation, and the fact that the thermostat is not having errors currently (I just tried it now the following morning when freezing outside and the reboot appears to have cleared away the previous issue), I do not plan to take the chance with a rewiring, so will for now abandon the objective of having an HA-controllable HVAC system, at least until I can set aside time to sufficiently and thoroughly analyze the alternatives.
I can, however, monitor both the blower and the heat pump electrical power consumption via my IoTaWatt, so that will have to suffice for now.
Thank you @CaptTom , @FredTheFrog , and @lordwizzard for your input!