I think you are back to where you started. If it was cheaper to use gas to run a heat pump then everybody would just run generators in their houses off of gas instead of using electrical lines.
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...
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: