We do this for gas. IMHO you end up paying monopoly rates for the pipes and then stupid game prices for the gas. Maybe the savvy consumer comes out ahead but seems like a net negative to me.
So assuming the pipe maintenance is done at cost, with no money not being spent on the network. What would your better net positive solution even look like?
But we need to build the nuclear reactors first.
In the mean time, no: people can’t just freeze in the dark.
If you do put in heat pumps, nuclear reactors are still one of the more expensive ways to heat a home, but you need a third as many of them as compared to the no-heat-pumps case, if you insist on heating only with nuclear power.
Nuclear power is really only important if you also want spicy atoms, because it's by far the cheapest source of spicy atoms. Annoyingly, this is now a thing a lot of countries have a solid reason to want.