I'm sure all those iai schools have their reasons for training drawing from seiza. Emphasis on training, as opposed to actually representing an actual situation that occurred (as in samurai in seiza with their katana still in their obi). It's not all from seiza though. There are iai schools with a sizable standing iai curriculum. Also, schools like Katori Shinto ryu have iai but none from seiza afaik.
You're probably right, I've only seen limited iaido and been through only one kata (although I was taught the kata by my (late) aikido sensei and observed the same kata 10 years later when an iai school was starting up in a different city. I wasn't around when people carried swords, I don't think any of my past lives have been in sword carrying countries, either - not katana, anyway. (and, no, I don't really believe in reincarnation)
Anyhoo... seiza/shikko/hanmi-handachi/etc are quite popular in Aikikai aikido as a way to train posture, hip strength, and to develop body movement. I try to limit the amount of it that I do largely because my knees have had a beating over the years - but after 17 years of aikido I don't recall my knees being worse than at the start - the skin was tougher, but no worse in terms of ligament, tendon, and cartilage... (and I'm not tiny - been knocking around 220 lb/100 kg for the last 10 years - not proud of that, but...)