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.