Here's how I break this issue down. I almost never record with omnidirectional microphones any more, but I used to do it a lot, and I still appreciate the qualities that they can offer.
Fact #1 is that the type and size of the microphone or capsule make fundamental differences. There are many good single-diaphragm omnidirectional condenser microphones with diameters of ~20-21 mm. There are also many that are larger, smaller, not condenser microphones, or that use dual-diaphragm capsules. They will behave quite differently, and should be considered separately. (I'm not knocking them, but there can be pitfalls if we breeze past certain implicit assumptions.)
Fact #2 is that in any normal, enclosed space, the farther you are from any sound source, the higher the proportion of reflected sound you will get in your ears or microphones, and the lower the proportion of direct sound will be.
Fact #3 is that most folks who make recordings know fact #2, but few realize the true extent of it, or the amazing variety of different paths that sound can travel between a source and your ears (or microphones), or the amazing amounts and kinds of unconscious processing that go on in our brains before we actually experience what we hear. People usually way underestimate how much our brains are doing to make speech or music "listenable."
Fact #4 is that the frequency response curves that are usually published for microphones show their on-axis response--but even if those curves are scrupulously honest and detailed, the way that pressure transducers of the usual size work (see Fact #1), that curve (or straight line) applies ONLY to sound arriving within a narrow range of angles in front of the microphone. For all other sound you'll get a significant reduction at high frequencies.
Thus the applications for omnidirectional microphones that have flat response on axis (e.g. the Schoeps MK 2) are really rather limited. They're beautiful at close range, but not on sound sources that extend across a wide angle. Spot miking a solo instrument or voice in a studio, fine. Otherwise, much of the direct sound will arrive off-axis--and if you back off the mike(s) in order to fit the sound sources into a narrow enough overall angle, then most of your sound will no longer be direct because of the distance involved.
Fact #5 (or maybe it should be Fact #0?) is that it is far easier to show why a certain type of microphone is inappropriate for a given application than it is to deduce from first principles what type of microphone ought to be used. A microphone or capsule that is designed to give flat response in a diffuse sound field (e.g. Schoeps MK 3, renamed a few years ago to MK 2 XS) won't necessarily be a satisfying choice, either, even in a highly reverberant space; it can be too edgy sounding (though maybe for some of the music that people here are recording, and some rooms and sound systems, that may be less of a risk than it is for the unamplified classical music that I generally record).
We don't always get a choice as to where we can place our microphones, but when we do, I'd say that most experienced recordists look for a position where there's a favorable balance between direct and reflected sound. Thus some kind of in-between response curve will be most likely to give you pleasing results most often, all other things being equal. I wouldn't start out with either extreme.
Fact (or editorial assertion) #6, before I sign off, is, don't be superstitious about equalization--all four Schoeps omni capsules, for example, have identical polar response, and a recording made with any of them can be equalized to be sonically indistinguishable from a recording made in the same mike position with any of the others.
And fact (or suggestion) #6-1/2 is to consider sphere attachments. They can make a very useful difference for not much money.
--best regards