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That is a critical find. What we have here is a classic "power stealing" issue that can be catastrophic for specific integrated furnace control (IFC) boards like the one in that DGAX070BDTA, DGAX056BDTA, DGAX090BDTA.
When a Nest thermostat is installed without a Common (C) wire, it attempts to charge its internal battery by "pulsing" or "robbing" power from the Heating (W) or Cooling (Y) circuits. It essentially closes the circuit just enough to get power without fully engaging the load.
This can happen with any power stealing thermostat, not just Nest.
Why it's "Taking Out" the Main Board
In the case of that Coleman/York DGAX unit, the combustion blower relay on the board isn't designed to handle that constant micro-voltage or rapid "chatter" from the power-stealing pulse.
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Relay Stress: The thermostats power-harvesting creates a low-voltage leak through the W terminal.
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The Failure: This leak puts the combustion blower relay into a high-resistance state or causes the contacts to "arc" and weld. Once that relay fails, the board is toast because the inducer will either run constantly or not at all, throwing a hard fault.
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The "Search": As you mentioned, the thermostat is actively searching for a path to ground. If it can't find a dedicated C wire, it uses the W circuit (the inducer/combustion path) as its ground, effectively using the relay coil as its power source.
The Fix: No Shortcuts
To prevent eating more boards on these units, there are really only two safe paths:
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Pull a New Wire: Always provide a dedicated C wire from the transformer to the thermostat.
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Nest Power Connector: If pulling a wire isn't possible, use the official Google Nest Power Connector (or a Venstar Add-A-Wire). This creates a safe path for that "stolen" current to return to the transformer without passing through the sensitive relays on the furnace board.
Tech Tip: If you see a Nest on one of these units and only 4 wires at the wall, it’s a ticking time bomb for that IFC board.
