I think "kickboxing" as a whole is kind of a generic term for strikers who use more than just the 2-point offense. But there are a few reasons why it (and western boxing) are showing up more than most other traditional striking arts. I'll offer my views considering Muay Thai and Boxing in general.
Generally speaking, when you're training in either MT or Boxing, you're going to be training with practical/live application in mind. Whereas a lot of Traditional Martial Arts may still hold on to 1-step or 2-step sparring in order to engrain form and execution under application, they are rather limited in an actual combat situation where things are chaotic and will most likely progress past the initial set of techniques (even when those techniques are successful). When training in boxing or MT, a large part of the training is implemented and sharpened during sparring when you have a live and resisting partner. Instead of the flow of training being a "Start/Stop/Reset" process, the sparring will be "Start...do not stop until this ends" continuing process that teaches one how to adjust, pace themselves, feint and/or setup your opponent. Further, both arts do well in teaching footwork and distancing, which is way more maleable (or applicable) on your feet than it would be on the ground.
Another point of view is this: As Mark Hatmaker would note (find his book "Savage Strikes") 100% of all fights start on your feet. 90% of all fights go to the ground. 90% of the time, the fighter that can take the fight to the ground will win the fight. Since all fights start on your feet, why not be equipped with the tools to do damage on your feet? Further, why not be equipped to defend against such standing attacks? It just makes sense. All MMA practitioners should be able to fight in all circumstances.
Further, the skilled striker always has a chance as the act of closing the distance for a shoot/takedown offers the opportunity to damage the person shooting when they are most vunerable. In this case, a knee to the head while shooting in can always be discouraging to any grappler or end a fight. You can see that here
when James Irving KOs Terry Martin with a flying knee as Martin goes for the takedown.
But this can be seen in stand up contests too, where great punches
can end a fight. I think where kickboxing has a larger application than just boxing is that leg kicks
have proven to be a serious weapon in the MMA world.
In the current stage of evolution that MMA is in, one must be able to fight standing and on the ground to be competitive. Most will be better at one thing than the other, but you have to be well versed in both. Randy Couture made "Dirty Boxing" a hallmark of his fight game as well as his wrestling. Genki Sudo has K-1 level striking to go with his phenomenal ground game. Bas Rutten was a phenomenal striker coming into Pancrase, and after he developed his ground game he became even more dangerous. As much as I love the ground game, effective striking can be just as lethal.