"Burning coal for heating is more efficient than electrical due to thermodynamical losses at coal power plant"

This is false, even though it is true that converting electrical power to heat would be even worse. The correct way to heat with a given amount of heat source is this:

• burn it at high temperature
• make work with heat machine between Thigh and the environment
• use the work to power an air conditioner to bring heat from Tenvironment into Troom.

This gives efficiency more than 1 (more heat brought into room than heat produced by burning), and net cooling of the surrounding environment.

"Burning coal for heating is more efficient than electrical due to thermodynamical losses at coal power plant"

This is false, even though it is true that converting electrical power to heat would be even worse. The correct way to heat with a given amount of heat source is this:

• burn it at high temperature
• make work with heat machine between T_high and the environment
• use the work to power an air conditioner to bring heat from T_environment into T_room.

This gives efficiency more than 1 (more heat brought into room than heat produced by burning), and net cooling of the surrounding environment.