Reducing temperature improves the thermodynamic efficiency of the engine, and combined with a higher TAS for the same IAS as at lower altitude makes it a better bet to fly higher.
The density argument is self-cancelling - yes, the mass of air through the engine is reduced as a result of increasing altitude, but so is the drag on the airframe, so thrust required is less.
Even at different altitudes, at the same IAS the dynamic pressure on the aircraft is the same; which is equivalent to saying that for a fixed geometry intake mass flow of air is the same. Therefore if IAS is constant, reducing density doesn't come into it. What does improve with altitude is TAS and thermodynamic efficiency.