Thermal power plants generate lots of waste heat that can be used for district heating. If you don't use that waste heat it will still be there. Worse if you need electricity for cooling you end up with even more waste heat.
Your assumption falls flat because a lot of people don't need to generate extra heat from electricity, they already have too much of it.
First off, AC is a heat pump, and heat pumps are always much more efficient than direct heating. A well designed AC system can be up to 500% efficient, since it only needs to move heat and not generate it.
Secondly, municipal heat is pretty rare in the US; few cities have it and the number keeps getting smaller each year. What few municipal heat systems that do exist are largely for business districts too, unless if you live in NYC as an American you most likely will be using either AC or forced air natural gas heat. Given the choice between the two[0], AC would be the best possible one from an efficiency perspective.
0: Heat pumps can be used to heat too, which is even better. Its my hope that heat pumps will become more common in the coming decades to assist in electrification.
> few cities have [municipal heat] and the number keeps getting smaller each year
I'm assuming by "municipal heat" you mean district heating. Can you explain why those systems keep getting rarer in the US? Over here in Europe, we are investing a significant amount of money in expanding those systems and connecting more houses.
Natural gas is dirt cheap in the US. Except in extremely dense urban environments with the physical plant already in place (NYC is the only one I can think of), it just would never pay back financially to use district heating. And if greenness/efficiency is the goal, using extremely efficient heat pumps and solar are probably a bigger win in most areas of the U.S.
My own district heat is also coming from a natural heat power plant, FWIW. I wager it's more efficient because they also produce electrical power (the heat is what remains in the steam after it's been run over the turbine to generate electricity).
Your assumption falls flat because a lot of people don't need to generate extra heat from electricity, they already have too much of it.