I'm in Yoshinkan as well and when I trained in Canada I don't believe anyone ever said "onegaishimasu" on the mat (although we always bowed out with "arigatou gozaimashita).
When I came to Japan my first experience with the word was to Kancho Sensei when we were introduced as the members of that years Senshusei course. I remember having a really hard time trying to remember how to pronounce it as I stood up in front of everyone and bowed to him <wry grin>. At the time it was explained to me as "please think upon me favourably" or something close to that that doesn't have an English translation.
Aside: If anyone's interested I think I was told to remember "On a guy she must" and drop the final "T". Okay..not politically correct, but I did remember it! Also, for "doitashimashita" or "you're welcome" I was told to remember "Don't touch my mustache".
During the senshusei course we used "onegaishimasu" as part of our morning shinkoku (report) to the instructors, but not on the mats.
As I stayed in Japan longer I started using it more because it is a large part of Japanese society and language. Interestingly enough, now that I am running my own dojo I have found that most Japanese beginners use "onegaishimasu" automatically when they are beginning to train with their partners. Yoshinkan has a formalized "bowing to partner" before beginning a technique and my Japanese beginners add the "onegaishimasu" during that. My foreign beginners just look a little stunned at the whole bowing thing (unless they've been in Japan for awhile).
I have also noticed that Yoshinkan children's classes have the kids yelling out "Onegaishimasu" at appropriate times. And, now that I think about it, my son's daycare events have the kids lining up and shouting "onegaishimasu" at appropriate times to each other and to the parents.
My son also says "onegai" meaning "please" when he wants something. And if I don't say yes immediately it becomes "onegai! onegai! onegai!".
I guess all this means is that I see "onegaishimasu" as being a significant part of Japanese society (like please/thank-you/excuse me) and is taught to kids as they grow up (like please/thank-you/excuse me) and since we're studying a Japanese martial art we're picking up on this stuff.
As to why its much more prevalent in Aikikai than Yoshinkan...I haven't a clue. Maybe it's because us Yoshinkan types use "Osu!" to cover a zillion meanings that include all those covered by "Onegaishimasu" and we wouldn't want to be inefficient in our use of language <grin>.