Well to start, the antenna generates it's own noise since it is not 100% efficient; i.e. it has a real resistance in addition to it's radiation resistance. If you put a real antenna in a box made of perfect electrical conductor, it would still generate noise, so you really don't halve the noise by increasing the gain 3 dB, you double the signal.
Here is a good tutorial on phased arrays. It's actually a lot easier to think of it when receiving a skewed wavefront, and the phase shifters (delay elements) compensating for the delay.
OK, while that seems like a valid tidbit if one has to implement this stuff, it doesn't seem like that's a helpful complication to introduce when trying to explain the basic concept.
Here is a good tutorial on phased arrays. It's actually a lot easier to think of it when receiving a skewed wavefront, and the phase shifters (delay elements) compensating for the delay.
http://www.radartutorial.eu/06.antennas/an14.en.html