I am not sure what you mean by school. Speaking Dutch myself, I know that school is also used to refer to what we call dojo (the place where we train) in contrast to school as a system (ryu).
Nevertheless the relationship between karate and iaido or kenjutsu fascinates me.
First there is the historical context that I mentioned before and which is different in Okinawa and Japan. In Okinawa, these days little reference to the sword can be observed (with the exception of Motobu Udun Ti, and with the shift in soke, this art moved also to mainland Japan). In Japan a number of very influential teachers were also trained in kendo or iaido. It was not uncommon to see demonstrations of sai or tonfa against sword by people like Sakagami or Hayashi). They might have incorporated some influences of sword tactics directly in their art but not much.
Second there is the individual search by Western karateka who through karate, hence exposure, get interested in other Japanese arts. I must admit that I belong to that category and I discovered traditional Japanese schools some 30 years ago when still very active in Karate and decided some years later to study them when I was in Japan (initially to train Shito-ryu with Mabuni sensei).
Several karateka moved on and started to become more and more interested in the Japanese sword. Also in the US, for instance Shimabukuro sensei (whom I met because of a mutual interest in Ono Ha Itto Ryu) and Demuro Sensei (although when he visited Belgium he told me he had a background in kendo and his teacher was Sakagami).
One of my first sword teachers (I trained Toyama-ryu in that time), Kurishima Sensei, was besides a great Iaido teacher also menkyo kaiden in Hontai Yoshin Ryu and had a history in itosu-ryu (he was ranked 5th dan).
My good friend Wout Verschueren was also a karateka but he got completely fascinated by Muso Jikiden Eishin-ryu.
One of the great things about training with weapons is a better understanding of ma-ai. Both the weapons in Hontai Yoshin Ryu as the kumidachi of Ono ha Itto Ryu are continually challenging my perception of distance and are training them in a different way than karate did (and no need to say that in order to be successful in karate kumite these concepts are also extremely important).
At least for me the transition to these koryu is a rich experience. I don’t argue that a thorough study of ryukyu kobudo is not an interesting complement to Karate (my good friend and old time karate buddy Alain Berckmans found interesting challenges through his teacher Nakamoto). My journey didn’t bring me there since I was taken away by my passion for Japanese koryu before I physically discovered Okinawa (that I visited only once).
What about iai-do, I still do it but my passion is with kumidachi.