The Carnot efficiency does drop as T_c increases, so all else being equal, for the same amount of fuel being burned, you'll get less energy. However, more importantly, at higher temperatures, a given volume of air at the same pressure contains less oxygen. All else being equal, at higher ambient temperatures, you're burning less fuel. For a turbocharged engine, you could increase boost to counter-act the reduced density, but that increases both mechanical and thermal stresses within the engine.
Anyway, I think the reduction of oxygen density is the primary driver for reduction in power as ambient temperatures rise.
The engine indeed makes less power, but since the air density is lower, you also have much less drag. (I know that for cars, the drag reduction dominates the power efficiency).
For airplanes take-off, the lift is the issue I'd say.
As a concrete example, current Formula 1 cars reach their fastest speed at the Mexico Grand Prix, held at Autódromo Hermanos Rodríguez in Mexico City. Current race record is 372.5km/h (231.5 mph)[1].
The elevation of the track is around 2200m, and as you say the loss of drag more than compensates for the loss of engine power. Though being turbo engines helps as I understand.
Yep it's lift and for some planes power loss due to less oxygen in the engine, but that highly depends on the kind of engine. You do get faster easier due to less drag but you also need to be faster until you have enough lift to take off. So maybe the takeoff run won't take that much longer, but you'll have covered a lot more distance.
Anyway, I think the reduction of oxygen density is the primary driver for reduction in power as ambient temperatures rise.