Bob Cain wrote:

Intuition tells me that the on axis response is the predominant
factor and that the angular variations from that are going to be
a very similar function across microphones of similar size [ ... ]

Two early morning thoughts:

[a] The response actually heard on a recording will be somewhere between
a microphone's on-axis response and its diffuse-field response, but how
close it is to the one curve or the other depends on the miking distance
and the recording environment.

Diffuse-field response usually ends up being more important to the sound
than most people seem to expect. Even in a close-up recording, more off-
axis sound energy is picked up, and is heard in the result, than most
people seem to think will occur. With moderately distant miking or in
classical music-style recording which can be truly distant, the on-axis
sound energy is a small fraction of the total.

[b] The capsule's design and construction has a huge effect on its polar
response at various frequencies, and this can't be predicted from its size
alone, especially in pressure gradient transducers.

The Neumann U 87 for example has a rather drastic (but reasonably smooth)
loss of high-frequency response off axis, while other microphones of
similar size may have bumpy, elevated high frequency response off-axis.
That matters a lot to the overall character of a mike's sound in a room.

--best regards