This just isn't the problem at all; it's obviously a problem, but not the constraint here. Fossil fuels let you store energy in a distributed fashion (e.g. cutting and stacking firewood for the winter), which is way better for handling peaks.
If too many people get burnt by oversubscribed grids in the winter, I would expect that the middle- and upper-class response will be propane auxiliary heating systems.
The only thing that using natural gas to burn for a furnace (vs. using it for a heat pump) gets you, is that you need more natural gas generation plants and the infrastructure to handle the additional power on the system. On the other side of the coin, you have all the natural gas infrastructure to each house premise to maintain... which can leak, have issues, etc etc etc.
Overall that seems like a reduction in total emissions, since you're using less natural gas overall to heat the same amount. Yes, it increases power demand, but is also lessens demand on natural gas by a greater amount.
This could be a overall net negative if the power mix is heavily polluting (coal... etc), but largely we're moving away from that over time and it will only get better.