Theory: How do thermostats without a "C" common wire get powered?
First things first: I'm happy to see that you, too, yearn for IP-based control of things without those things caving in to the pressure to cloudify. Oh that more devices would offer a cloudless local network only option, and leave it to the user to implement VPN or other means for remote operation.

Anyway, in days gone by, thermostats were simple switches. It could literally be a toggle switch you'd manually throw to turn the heat on and off, or it could be an automatically-operated switch where a bimetal element or spring caused the switch to open and close based on temperature. Things were simple. The switch completed the circuit and the transformer's voltage was applied to the coil of a relay. Then came electronics.

Battery-powered thermostats can play a variety of tricks to extend the life of the batteries. One is the use of latching relays. One applies a current only to change the relay's state; it then keeps that state forever with no power consumed in the coil. Another current through the coil can cause the relay to change state again. Another trick is to steal power from the control circuit. As you noted, some amount of current must flow in order to make the relay in the furnace activate. A power stealing thermostat can let some nominal current flow -- not enough to activate the furnace relays -- and use this power for part or all of its own operation. This might be accomplished with an input current limited switching power supply of some sort.

It sounds like you have enough electronics background that you'll be able to sort out what's going on with the existing equipment if you're so inclined. It's possible that your zone controller may have failed, but based on what you've said of the system, I won't be surprised if it and/or the dampers and other equipment just never has been correctly installed/configured.