Tom, can you clarify on how to take the average height of the pattern?

It is not the average height. It is the average reception power.
Suppose you change the height of RX antenna (preferably together with
TX antenna) by (more than) a half wave length, you'll observe maximum
level Px and minimum level Pn.
So the reception voltages Vx and Vm are sqrt(Px) and sqrt(Pm).
The average voltage Va is then (Vx+Vm)/2. Thus average RX power is Va^2.

Also, if I keep the TX and RX antenna close, wouldn't that eliminate
the far-field effects?

It is not a far-field measurement. But the error might not be as
much as 3dB if the distance is such as two to three wave-lengths,
which is sometimes better than the bad average measurement performance
or doing nothing.

Tom/jj3fbs