I think it hits the nerve differently, so some of the conditioning doesn't apply.
If you have never gotten kicked before, your untrained nervous system thinks it is being killed, and freaks out. "Being conditioned," is half just teaching your nerves that being kicked isn't that big of a deal, and you are ok. If someone kicks you differently, it will hurt in a similar way as if you had not been kicked at all.
I also think that pushing off with the shin after landing a Muay Thai style kick makes it harder for the opponent to either kick you back or catch the kick. If you are used to the snapping style kick, it doesn't do anything to off balance you because it only causes trauma to the muscle where it hit. So if you aren't put off by the pain, you can kick back right away. And that's all people do in Kyokushin: stand in front of one another like rockem sockem robots trading body blows until someone flinches from the pain, and then they can throw a falling / jumping wheel kick, knowing that if they miss, the ref will protect them and stand them back up.
I think kyokushin kicking is fine, but it isn't the end all be all. If you are used to their kicks, you can pretty much just check them and throw straight punches at their face. Usually they can't grapple without grabbing your shirt, and they aren't used to parrying.
Look at how the kicker just stands there, straight as an arrow, putting his weight into the kick. Hands down. Head online. Slow to return to stance because he used the snap instead of the push. He couldn't do that in a regular kick boxing match, which is why in free style you see the snap kicks to the head more, instead of to the body or leg.
