Suppose you did pay $300 on one cold night... but it also turned out that your electric bill for May and September was basically $0. Would that be the worst thing in the world? Utilities are monopolies, and regulators keep an eye on their total bottom line. A massive yearly profit would come in for scrutiny.

Rather than maintaining individual gas heaters, it's more likely that the energy companies would maintain gas-powered peaker plants. If they operate 1% of the year, then we've cut 99% of the carbon emissions, with no loss of service. Even if they somehow had to operate 10% of the year, that's a 90% drop in CO2.