"high efficiency" turbines are not all that efficient. Burning gas releases 100% of the available energy as heat. Converting that gas to electricity is << 50% efficient. You also lose ~5% just transmitting the electricity to homes.
Also, electricity from gas is relatively expensive. Other sources like coal or hydro are cheaper, which lowers the average cost of electricity.
If you combine gas turbines with district heating you can recover almost all of the heat energy. If you use the produced electricity for heat pumps you come out ahead. For climate change purposes it is way better since gas pipes to homes leak way more methane than those to central power stations
There's not really a "should be" in thermodynamics. All heat engine cycles have upper bounds of theoretical efficiency, and burning gas in a gas turbine to generate electricity to create resistive heat is never ever going to be more efficient than burning that gas at the point you need the heat. It's simply not possible. There's always going to be losses - the exhaust gas will contain energy, there will be mechanical and transmission losses. There is no physical way we can change that.
The best thermal power plants - combined cycle gas turbines - get about 60% efficiency. Most are closer to 45%.
What we can do is either use the excess heat from the gas turbine in district heating (which combined with resistive electrical heat probably approaches the same efficiency as a local gas boiler), or use that electricity to drive a heat pump, which gives a greater than 1x return on heat where you want it. An efficient combined cycle gas turbine driving a heat pump is going to give you more heating than the same gas being burned in a boiler. - 0.5 * 4 = 200% efficient.
You can also get (although they're much less common) gas powered heat pumps (eg propane fridges in RVs). They might have a "primary energy ratio" of 1.5-2, bringing the total system efficiency of a local gas powered heating system back up to pretty close to that of a remote generation + electrical heat pump system. The electrical system has the benefit that you can slot renewables into the mix as well.
Also, electricity from gas is relatively expensive. Other sources like coal or hydro are cheaper, which lowers the average cost of electricity.