> Electricity generation efficiency from coal is only 25-50% according to [1].
25% efficient generation is from ancient garbage that has largely already been retired because it's uneconomical.
Combined cycle turbines have efficiency up to 64%:
https://en.wikipedia.org/wiki/Combined_cycle_power_plant
Power plants in active service are probably around 50% efficient.
> Heating efficiency of modern home furnaces is typically way above 90% according to [2].
That number is for the newest highest efficiency furnaces. By comparison you could say that commercial power generation with cogeneration achieves even higher efficiencies, but neither is widely deployed.
Existing furnaces are as low as 56% efficient:
https://www.energy.gov/energysaver/furnaces-and-boilers
The average for the existing installed base is probably around 80%.
Someone evaluating this for personal use might research the efficiency of their own furnace or compare the cost of installing a more efficient one vs. the net cost (possibly net profit) and environmental benefit of heating with electricity generated by rooftop solar.
> I'm treating natural gas the same as coal since, while the direct emissions are about half the CO2 per thermal unit produced, its production is much more problematic (esp. emission of methane) [4].
Then you have to count the same thing the other way when you're heating with natural gas. Fuel oil is even worse because it emits more CO2 per unit heat than natural gas and is extracted using the same methane-leaking process. Worse than that, many oil wells capture only the liquid fuel, have no infrastructure for capturing the natural gas and knowingly release it directly to the atmosphere!
Assuming your equivalence between coal and natural gas, that commercial power generation and transmission are 47% efficient (50% x 94%) and home furnaces are 80%, the percent non-fossil fuels needed for breakeven with natural gas heat is 41.25%. Some even lower number for oil heat. That's pretty much exactly the existing US average (and thus already ahead in places like California which are at >51%), and is only going to move in one direction as fossil fuels get phased out of the power grid.