all we need is a "24 volt common wire" not a ground. The old thermostat did not need a 24v complete circuit to thermostat as it was battery powered.
Here is how it works,
Your transformer sends 24 volts out through 2 wires (HOT and COMMON).The common goes to all 24 v relays (fan, gas valve, outdoor contactor etc) at all times.
The HOT goes to R at thermostat
Thermostat is basically a "light switch" opening and closing circuits.
On call for HEAT R and W closes the circuit to gas valve
On call for AC , R ,G and Y should all close sending circuit to blower and outdoor unit
On call for FAN R and G should close the circuit to blower
Now the new thermostat just like relays ,contactor. etc requires a COMMON.
Can you run anew wire from thermostat to the unit?