(I rewrote this post because some people complained that it was obscure. I hope this version is better.)
I think that two things are getting confused here. Nearly all large-diaphragm microphones have dual-diaphragm capsules, and nearly all dual-diaphragm capsules have large diaphragms. A few types of microphone have been made (and some are still) with large, single diaphragms or with small, dual diaphragms, but they are definite exceptions. As a result many people confuse the categories, and attribute sonic qualities to large-diaphragm microphones that are actually due to the dual-diaphragm capsule construction which most LDCs have.
Since so much recording is done with cardioid microphones, I'll focus on them. Dual-diaphragm cardioids all lose directivity substantially at low frequencies; their pickup patterns become more like a "wide cardioid." Please see the attached polar diagram for the leading microphone of this type, the Neumann U 87; its low frequency curves are on the left. Notice how they "bloom outward" at the sides and back as the frequency decreases. With the older, shared-backplate type of dual-diaphragm capsule such as the one in the Neumann U 47, this loss of directivity at low frequencies is even more pronounced.
This is one reason cardioid LDCs are the traditional choice for close-miking of vocal soloists in studios: Not only does proximity effect give them a bass and mid-bass boost for front-arriving sound, but the deterioration of their pattern at low frequencies causes them to pick up extra room sound at those frequencies. So they sound "rounder" and "fuller" and "warmer" than if they were true cardioids all across the spectrum. But that's not because of their diaphragm size; it's because of their dual-diaphragm construction. Small dual-diaphragm cardioids have similar low-frequency characteristics.
Here's where I may have to ruffle a few feathers, which I hate to do (cough, cough): The above-mentioned low-frequency pattern characteristic turns into a real disadvantage for recording in stereo with coincident or closely-spaced dual-diaphragm cardioids. The widening of the directional pattern reduces the difference (increases the correlation) between the channels. The low frequency part of the direct sound tends toward imaging in the center in playback--it becomes practically mono. Back in the LP era that was considered a good thing, since any difference between left and right channels translated into vertical groove modulation, and too much vertical modulation at low frequencies would lead to skipping. But difference information at low frequencies is also what gives a stereo recording a sense of spaciousness, so this is one of the areas in which digital recording is greatly superior to LPs.
To my way of thinking, coincident or closely-spaced stereo miking calls for directional patterns that are as uniform as possible across the audio frequency range--or alternatively, if the directional pattern of a cardioid could become narrower at low frequencies, that would produce more difference between the channels where it's needed. (Such patterns can be obtained with Soundfield microphones or with the Schoeps "Polarflex" system; the latter also allows for space between the left and right microphones of the pair). But using coincident, dual-diaphragm cardioids to record music in stereo is generally a rather bad idea, and it is only made worse when the capsules have large diaphragms.
--best regards
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