Is there a reason heat pumps use electricity? I would have thought the same approach would work with gas - you only need to burn a fraction of the gas to drive the "fridge" backwards?
I don't think that's right: look at micro-CHP (Combined Heat and Power) systems: they run an engine to generate electricity, and then capture the heat for heating. I don't think you can get them for residential in the US though.
Well I found this, they used the heat for hot water:
Except it's not legal to do this, and even if it were there'd be a lot of hassle.
If you plumb the radiator to your home you get >100% efficiency
[1] https://www.ehs.harvard.edu/sites/default/files/emergency_ge...
[2] https://www.mass.gov/doc/310-cmr-700-air-pollution-control-r...