exactly Andreas, and that is why you normally optimize receive antennas for G/T or Gain over (system) noise temperature. Because this will end up in Signal-to-noise ratio.
Having a huge G with crappy T might be worse than decent G with low T. So depending on your setup, a slightly underilluminating feed will produce better results as it won't pick up as much noise from behind the main reflector (as for me, being close to the wall).