Heat pumps are significantly more efficient than 100%. They can get to 500% efficiency. So no, it's definitely not more efficient to burn gas in a home. (To say nothing of the safety of running gas lines to every house.)
You said
> Burning it for heat in the home is much more efficient than burning it in a plant, converting it to electricity, transferring that electricity, then turning that electricity into heat.
Burning it for heat in the home may be something like: 99% transport efficiency (gas distribution systems lose maybe 1%) * 80% combustion efficiency (a lot of heat energy still goes out your chimney as exhaust). Call it 79%. If you spent more for a high-efficiency burner with extra heat-recovery stages, you could get into the 90's.
Compare to: burning it in a plant (power plants can run efficient combined-cycle infrastructure, which is about 60% efficient turning it into electricity), transferring that electricity (plant-to-home transmission & distribution losses are 8-15%, so call it 85%), then turning that electricity into heat (and here is where heat pumps shine... heat pumps don't burn electricity, they use it to move heat, so they can have efficiencies above 100%).
So compare that 79%-90% "burn for heat in the home" efficiency to 60% * 85% * 300-500% = 150-255% efficiency for "burn it in a plant, convert to electricity, transfer that electricity, then turn that electricity into heat".
It's just not the point I was actually responding to. The OP was under the impression that you would be "extracting a lot more energy" from the gas in a plant vs the home.
Yes, converting gas to electricity allows you to use a heat pump, which in many cases uses much _less_ energy than a furnace to heat a home. (As long as it's not too cold.)
But that's just not what I was talking about. I was talking about the claim of extracting _more_ energy from the gas. Good day!