And even if you don't, a 60% efficient gas turbine at the power-plant, paired with a 3+ COP heat pump in the home will be 2x more efficient than burning gas directly in the home.

Using gas in an adsorption heat-pump is quite inefficient and unlikely to break even economically. Perhaps a gas turbine home boiler could be produced, that combines a mechanical heat-pump with a high efficiency combined cycle turbine; alas, they do not exist and would be very expensive. It's a startup idea.

A home gas furnace can easily surpass 90% efficiency. Turning the gas into electricity and then including the transmission losses to the home is going to be way below 84% efficiency. Plus reusing the heat produced at the power plant is much more difficult than reusing the heat produced in the home, when the purpose is heating the home in the first place.

Up to 64% as stated in the article. The 84% efficiency statement is in reference to the theoretical efficiency of a Carnot cycle.