A/C for cooling is one of those things that highly correlates with the availability of solar power.
So your assumption that they are running on coal is suspect.
Heatpumps running in the winter are a more pressing concern, since those are highly correlated with the unavailability of solar and therefore often run on coal or gas (unless wind or hydro is available)
Even so, burning gas in a power plant and then running a heat pump is more efficient than burning gas to warm a house.
Yes, ideally all AC will be running off solar, but that’s not the case at the moment. My state has thankfully closed its last coal powerplant, but also shut down one of its nuclear plants, using gas to replace both. We are now running at 50% gas 20% nuclear 20% hydro and 10% wind/solar. Which is why I wanted to focus on methane in this specific calculation: when deciding “is it OK for me to run the AC now, or is the longterm global heating side-effect too great?” natural gas is what is relevant to me.
How “great” that is is precisely the question here, and apparently it’s 2.2x. If you are really a stickler for exact real-life electricity production piechart distribution, multiply that by 50% gas and call it 1.1x. That is, for every year that I run my 1kW AC, that’s as if I am airdropping a 1.1kW heater to a random location on Earth that will heat it up at 1.1kW forever. 10 years = 11 random heaters. 8 billion people = 88 billion random heaters. Is that “too great”? I dunno.
Winter heating is its own problem, but at least cold can always be dealt with by more insulation and clothing. Heat can literally make whole areas of Earth unsurvivable without electrical cooling. Would I rather feel more comfortable now or choose to be able to survive without mechanical aids later?
An A/C consumes more energy when the temperature difference is higher, which is when it’s sunny outside. At those points in time, the grid is receiving a lot of solar power.
So just saying a grid has 10% solar is too simplistic. That grid probably has 30% solar during summer noon and 0% solar on a winter morning.
If your goal is to save emissions, your best bet is to get some solar panels if you can, run the A/C when the sun is shining. Have a well insulated house that acts as a thermal battery and turn the A/C off during the peaks of the duck curve.