If you want to train the most advanced grappling you can, you should always go to a specialist teacher.
However, not everyone can do that, not everyone WANT to look outside of their base art, not everyone wants to learn more than the basics of grappling.
Karate has at its base some rudimentary grappling skills. They fell into disuse when Karate was exported to the Japanese mainland, but it is there if you look for it. Kyokushin has a certain influence from Aikijutsu and judo. It fell into disuse as sparring and competition became popular and took up more time, but it is there if you look for it.
PS. Polarbearfighter. I too did jujutsu kai (and it is not a trad jujutsu style. It is a recreation from Judo, aikido and some point karate). I took it for a few years side by side with karate. I more or less got the same impression as you, but the skills taught me in jujutsu has been invaluable when trying to make sense of a lot of techniques and moves in kyokushin that just never made realistic sense as striking/blocking techniques -even when they are called striking/blocking techniques. I especially found throwing techniques all over in kata´s. That may be one of the reason I advocate bunkai so much.
I cringe when I see a obvious throw described as a urakento the chest (supposedly strong enough to break bones? Really?) followed by TURNING AWAY from the opponent and punch straight up into the air behind your shoulder (supposedly hitting his chin)followed by a turning jump over a lowkick landing in a crounching stance Xblock (pinan Go). It is a unrealistic overcomplicated explanation for what is obviously a solo representation of a ippon seoinage.
Sorry for ranting on.
Point is, karate has rudimentary skills, and it is a crying shame we do not teach those more often -if only to keep the traditions alive. But if you want to specialize, you need to go to as specialist style dojo.