Personally, I don't think that it's really a tradition - it's a combination of smooth canvas mats (which encouraged the practice) and young students (whose knees could hold up to the training) after the war.
The young guys on smooth mats introduced a lot more movement than existed previously. If you look at Daito-ryu and other traditional schools there was very little of the moving around on your knees that you see in Aikido today.
Yasuo Kobayashi used to say that he was the one who introduced shikko as an exercise - in the 1950's.
People today are paying the price with their knees.
Very interesting and informative post Chris, thanks. The last time I saw Kobayashi sensei was when he visited my dojo back in the mid to late 90s. I still remember being impressed how a person of his age could move so smoothly and freely in suwari waza, as if he had rollers on his knees and feet! He seemed to enjoy that training so much that it gives weight to his claim.