If we really cared about the warming component of climate change, we would stop funding natural gas and dump all the money we would have spent on natural gas infrastructure on renewables. We would roll back the ship fuel sulfur ban (which has a cooling effect) and rely on corrective measures for pollution problems in exchange for preventative measures for warming problems.
Because of methane leakage and other factors, natural gas is only a net-positive effect over coal in terms of greenhouse effect somewhere between 20 to 100 years down the line (imo reasonable estimates put this at around 40-50 years). Until then, natural gas has a more significant warming effect than coal because of obscenely high rates of methane leakage during transmission and distribution (which are underreported by relevant agencies to the order of 2-5x).
There is no one solution to climate change, but the question now is whether we care about it as the only problem or just one of a myriad of problems caused by human activity.
I don't think you understand thermodynamics.
ACs are heat pumps that use temperature differentials to move heat from one side to another. There are inherent losses there (e.g., moving 1000kJ of heat out of a room might take 500kJ, for 200% efficiency). That excess 500kJ is dumped outside into the world along with the 1000kJvof heat, creating a local heat island effect. That's why ACs consume electricity, and that's where the energy goes.
The radiator behind the freezer isn't mega hot because of advances in insulation that limit the amount of heat that needs to be moved and advances in efficiency when operating in specific temperature regimes. A modern fridge consumes 400kWh a year, which averages out to 1.1kWh/day, or 45W continuous draw. That's about the same as a laptop charger. But, well, obviously your house is much larger than your fridge. A fridge might average 400L in volume, but your house averages more like 600000L (1500x more).
If you could move heat around without incurring losses, you could use that to construct a perpetual motion machine. Conservation of energy is a thing and entropy always increases.