Multi-coating allows greater light transmission.
I know that there is flare even between lens groups and lens elements that have air surfaces, however even with a lens shade that is masked to the exact proportion of the image, and is at the borders of the image, there is still a certain amount of off axis light entering the lens. Wouldn't an MC filter cut down some of this light, and therefore reduce some of the flare?
Uncoated and single-coated lenses suffer from more "veiling glare" than do multi-coated lenses. This glare acts as a veil, obscuring low-contrast details. Other posters have explained exactly what causes this - light ricocheting between various elements - and even inside the glass lens elements themselves. A single-element lens thus benefits from multicoating, too.
If the lens is multi-coated, more of the light goes right on through - to the film. Thus, there is less light reflecting and refracting back and forth between various air-to-glass surfaces. Even cemented glass surfaces will create some [but very little] of this stray - or uncontrolled - light, bouncing around.
Multicoating a filter accomplishes two things: {one} more light goes through to the next glass surface, and eventually to the film. {two} because there is less light lost at the surfaces, there is less light bouncing back and forth between the front and back of the filter's glass surfaces. Eventually, this bouncing light exits the glass, and it doesn't go in the same direction as the light that originated from the subject. Thus, it "veils" the image on the film by landing where this light should not be.
A multicoated filter will perform better than a less-well coated filter. No filter will reduce the veiling glare inherent in the lens. Also, every multicoated filter will introduce more veiling glare than using no filter.
In old movies [and photos] you will see huge reflections off eyeglass lenses. Often the person's eyes are not visible at all. There were no coatings then. Now, eyeglass lenses are nearly invisible as very little light reflects off the front, or back surfaces.
In the film "Good Night and Good Luck", I suspect the eyeglass lenses were purposely uncoated, to give that "period" look.
Multicoating allows about 99.5% transmission at each air-to-glass surface. Uncoated glass transmits about 96%.