That's why this could work - the panel is the logical place to insert the DC supply. If ground and neutral were bonded in multiple places then it would just be a DC short.
The reason is that you don’t want to allow a fault which puts a line voltage (120VAC) onto the chassis of a piece of equipment waiting for a human to come along and complete the circuit to ground. So, you use the protective earth to connect to the chassis, such that if a power line ever did contact it, it would be a dead short to ground, which would trip the OCPD (over-current protection device, typically a circuit breaker), clearing the fault.
If you replace this safety mechanism with something that’s high-enough impedance to maintain a 12VDC differential, you’ve eliminated this important safety mechanism in your AC distribution system.
If there was also a DC device on the same side of the break as the AC device, then you'd have a possible return path for the AC current that goes through the DC device, the ground wire, and the DC power supply.
In addition to that, would it also cause problems with GFCI?
The 12 VDC return current would be on the neutral. I'd guess that a steady DC current on the neutral wouldn't cause problems, but whenever a DC device is switched on or off, or you use a DC device that has a fluctuating instantaneous power need, that should induce current in the GFCIs sensing coil that won't be canceled out by an opposite current induced from the load wire.
I don't know if that mismatch would be long enough or large enough to trip the GFCI.