Different constraints. Better cooling allows the processor to throttle less and achieve higher average performance, which incidentally would also reduce your battery life if you had the processor running at 100% load the whole time.

Higher performance finishes the race to idle quicker and thus counterintuitively saves battery.

not at all: higher performance equals higher freq, which requires higher voltage. Power scales w/ frequency linearly and the with the voltage squared. Then any active cooling will have its own power consumption.

Best battery/power savings are achieved running as low as possible voltage for as high as possible frequency. At such voltages the CPUs/APUs are less likely to throttle to begin with.