OK, "omnidirectional" is an absolute adjective, like "in order to form a more perfect union" in the preamp to the Constitution, so you have to give the writer a tiny bit of slack.
The issue is that the wavelengths of sound at the highest audible frequencies are very short. Long waves can flow around solid objects, but shorter waves tend to be reflected or bent (diffracted) depending on the shapes and dimensions involved. Generally if a solid object is half the wavelength of sound at a given frequency, its presence in the sound field will disturb the sound at that frequency (and higher up).
So if you do the arithmetic, if sound travels at 1100 feet per second, then 1 inch is one wavelength for around 13,200 Hz, so it's a half-wavelength for around half of that, or about 6600 Hz. And if you look at the polar diagrams for microphones with a diameter in this class, you'll see that the pattern starts to become narrower right around that range of frequencies. Even though it is still a pressure transducer, the pattern is no longer omnidirectional at the highest frequencies.
That's the appeal of smaller (e.g. 1/2" or 1/4") diameter omni capsules: they retain their omnidirectional pattern to correspondingly higher frequencies. That's important in certain acoustical measurement applications. Whether it matters so much for audio recording is another question entirely--generations of engineers have used microphones which were 1" across or even larger, despite the fact that their polar pattern at the highest frequencies was far from ideal. This is because the sound arriving on axis is mostly direct sound, while the reflected sound in a hall mostly arrives off-axis, and you generally want some high-frequency absorption to take place for indirectly arriving sound, so that it doesn't smear the stereo image.
But it all depends on the room acoustics, the miking distances, the type of music, and most of all, what you want out of a recording, really.
Generally I find most useful a pressure microphone which is some compromise between a free-field and a diffuse-field type. I have a pair of free-field capsules (Schoeps MK 2) and have used them exactly three times that I can recall in 30 years of owning them. I also own a pair of diffuse-field microphones (Neumann KM 83) which I used to use a lot back in the mid-1970s, but those recordings sounded pinched and metallic. Later on I learned how to equalize them to make them sound a whole lot more natural--I've had a good time reclaiming those old recordings lately.
I don't record with spaced omnis all that terribly often any more, but when I do, I usually use a capsule type which was specifically designed for placement near the distance at which the direct sound and reverberant sound in a hall approximately match or balance one another. They're from Schoeps (MK 2S), so sue me.
But I'm not endorsing them above other compromises of similar type and quality, which absolutely do exist in the world--frankly the easiest type of condenser microphone to make well is a pressure transducer. Not many people make a good supercardioid or figure-8, but there are plenty of pretty good and even really good omni condensers, and even some dynamic omnis that are worth listening to.
--best regards