How does the use of a reflector affect the performance of a receive antenna?

[Baluncore]

The feed is just one means of creating, so far as possible, a uniform amplitude and phase across the aperture.

That is the last thing I would want. By tapering the illumination to the edge of the dish, the side lobes can be significantly reduced while only slightly broadening the beam.

The slope of a parabola varies in proportion to the radius, the angle is Atan( slope ). The illumination, or aperture distribution, can be calculated from the geometry of the dish and the polar radiation pattern of the focal element. Once the aperture distribution is known, the beam pattern of the combination is the 2D Fourier transform of the 2D aperture distribution. Any step in illumination due to the edge of the reflector will result in ringing in the frequency domain. That makes the big side lobes.

But none of that detracts from the fact that the energy scattered by a “paraboloid” receive antenna is determined only by the focal transducer configuration, and not by the presence of the reflector dish.
The direction of scattered energy will be determined by the presence of the dish, but not the total energy scattered, nor the efficiency of the receive system.

The energy reaching the receiver can be doubled by replacing a bare dipole at the focus with a more complex antenna such as a reflector backed dipole, a yagi or a horn. +3 dB makes a big difference near the noise floor. That explains why radio astronomy dishes do not have a simple dipole at the focus. The reciprocal, or transmit analogy, is that a reflector backed dipole as a driven element will double the energy illuminating the dish surface compared with a simple dipole.