> how does any sort of flat plate collect incoming radio waves as well as or better than a semi-spherical (?) dish?

So the really counterintuitive fact here is that the amount of far field energy that an antenna can collect is a tradeoff between how directive it is and how much it can collect from a particular direction.

i.e. assuming your antenna is 100% efficient, and say antenna A collects energy from all directions, and say antenna B collects energy only from the "top" hemisphere, antenna B collects energy from that direction twice as much as antenna A. In more technical terms, the directivity integral is always constant.

What's also kind of wacky is that generally speaking the bigger the antenna is, the more directive the antenna is--e.g. an antenna that's big generally collects better from its "boresight". The first paragraph still applies, so a big antenna is poor at collecting energy off-boresight.

A phased array is a curious idea. Start with a passive phased array, where you have one antenna element, and you add the contributions from that one antenna element copy-pasted in space.

generally speaking the further an antenna element is spaced apart, the more directivity you get, since the physical size is larger. If you have many of these elements copied across a large surface, your directivity increases a lot.

A passive array is kind of dumb since you might as well just have a parabolic reflector antenna ("dish")--but an active array is where it shines.

You can tune the exact phase and amplitude contribution from each phased array element, and when you do that, you can tune the exact radiation pattern of the phased array. You can steer the "boresight" by delaying the input from some elements; you can make the antenna less directive or more directive; you can make it so that there are multiple boresights (useful if you want to track multiple radar targets); etc.

For pure efficiency and directivity, a parabolic reflector is best. That's why it's used in radio astronomy. Active phased arrays are used in military and high performance radios since you can change the radiation pattern pretty much instantaneously.