Re: "Kickboxing comes from karate": That's only partially accurate. (That means it's also partially inaccurate.)
In North America, kickboxing essentially happened as a hybrid of full contact "sport" karate and boxing. Practitioners wanted real contact; not point fighting. And that's why you generally don't see the use of knees or lowkicks in "American" kickboxing; because its mostly derived from styles of karate that generally don't deal in knee strikes and low kicks. (American style kickboxing = American sport karate + boxing.)
In Asia,
kickboxing already existed in the form of a number of South East Asian fighting arts. Japan got onboard when some karate fighters (especially Oyama / kyokushin fighters but other styles as well) wanted to get involved in full-contact fighting. They based themselves on close observations of and close collaboration with muay thai fighters. They essentially adapted those techniques and approaches to their karate base. This is why pro kickboxing outside of North America has generally included knee strikes, low kicks, clinching, etc. They might not do headbutts and elbows, but the rest is evidently inspired by the fighting arts of South East Asia. (Japanese kickboxing = knockdown karate (AKA karate that is heavily influenced by Muay Thai) + head punches.) This is why Japanese kickboxing (and kickboxing throughout most of the rest of the world) includes knees, low kicks, clinching, etc. And this is also why pure knockdown karate practitioners have historically had to go to "kickboxing school" before hitting the K-1 stage. It's well fine and dandy that they had their black belts in Seidokaikan or Kyokushin; they nonetheless needed to get additional training on protecting their faces from punches, and on adapting what they've learned for a different set of rules and conditions.
https://en.wikipedia.org/wiki/Kickboxing