Technically, the Korean language definition of "Kihap" has nothing to do with either yelling or breathing.
Anyone ever hear of Hapkido? 합기도
합 = Hap - meaning coordinated as in merger; mutual; fusion; union
기 = Ki - meaning high spirits; energy; vitality
합기 (Hapki) = Coordinated energy
기합 (Kihap) = Energy that is coordinated or focused
It is a coordination of mental focus and physical energy. It is defined in Dong-A's Korean/English dictionary as "will power."
In this context, "Kihap" is a noun - - the "will power" or "focused energy" that you possess at any giving time. You can increase it, or decrease it at will depending on your mental focus. Thus, any Taekwondo technique can be performed with "Kihap" (a blend of intense spiritual energy/will power with physical action), and never make a sound. To increase physical assertion and power, proper breathing is naturally one of the many methods, therefore breathing or breath control is often associated with the moment of intense focus, or "Kihap," but by original definition, it refers neither to breathing nor yelling.
On the other hand, in the context of Taekwondo, as a "yell" or "shout with energy," the term Kihap is being used as a verb. This might not be a common language usage throughout Korea, but it is a well established alternate definition as used in the Korean Martial Art community. Professional jargon is often adopted by people who define terms according to a specific field or skill. Personally, I wouldn't refer to Kihap (meaning a yell) as being an elementary school definition, since it has been used for decades by adult Taekwondo Masters conveying a concept to adult students. Countless times, I have heard native born Korean speaking Grandmaster say, "Let me hear you KIHAP!" or "Kihap louder." Not only does this imply an action verb rather than a noun, it specifically refers to making a sound.
If a term like "noogie" was commonly used to refer to a childish playground activity of putting someone in a headlock, and rubbing their head with your knuckles, then that's one definition. If Professional Wrestlers, or even combat fighters, started using the same term as another name for a "pile-driver" where you turn a person upside-down and drive their head into the ground, potentially causing serious head injury, broken neck, and death, then I would say that the nature of the term "noogie" would change, and would no longer be an elementary school one. I believe that the term "Kihap" as yell has been used by skilled adults and professional instructors who have a clear understanding of the language, and its meaning so as to validate this as a legitimate adult definition for modern usage.
Can we perform successful, powerful strikes without an audible "Kihap?" Yes. Does a louder Kihap help to increase the power of a strike? This might be debatable in some people's minds, but not in my mind. Why do power lifters make such an audible yell when lifting heavy weights. Have you ever seen one of them win an Olympic Gold, or break a world record while lifting silently? Women giving natural childbirth don't just "push" quietly. Some of the yell off-sets the pain, perhaps helps to release natural endorphins and adrenaline, but the more intense contraction of internal organs is facilitated by the yell. In addition, it helps to keep us from NOT breathing correctly. Military soldiers chant when they run to encourage proper breathing. A weight lifter who does not breathe during a bench press, might just turn a few shades of red, then pass-out.
I agree with DArnold and the others who say that the power of "Kihap" (as a noun) comes from a coordination of physical and spiritual (or even mental focus), and while this type of focused "Kihap" can be used on every technique, I would reserve the maximum Kihap "yell" (as a verb) for finishing maneuvers, or those times that I wish to distract or destroy my opponent without a touch. In my own personal experience and belief, the audible Kihap
does increase power significantly and if I am seeking maximum damage to a target, I will use it. Of course, this is just my personal opinion!