There's also one more fact that I think other commentators glanced over. One of the many approaches you can have towards gain and directivity of an antenna is the idea of an effective cross section of an antenna.

Think about a simple dipole. It basically has no cross sectional area, but yet it is still capable of capturing free air propagation and confining it to a conducted EM wave (into coax, for example).

This is because the presence of the dipole creates a disturbance in the way free air propagating EM waves (which can be approximated greatly since it's in free air), and creates more complex "near field" phenomena which essentially increases the cross sectional area of the antenna.

In this sense, all antennas disturb the free air propagation and create near field cross sectional areas that it will use to capture EM energy and send it through a conductor.

Indeed, there's actually a relationship between directivity/gain and this imaginary cross sectional area. I haven't done this in a while but I believe for parabolic antennas (or any antenna where the physical size >> wavelength) this cross sectional area is essentially the same as the physical size, and hence this is one way to think about the high gain of parabolic antennas like the ones used for radio astronomy.

In that sense, in certain degenerate scenarios, the flat plate might have a larger cross sectional area than an actual semi-spherical dish (hint: it has to do with physical size vs wavelength of the wave you're dealing with)