My very first Aikido class, I was slightly late for (what fool builds two community centres next to each other?) And that was the first thing I was struck by, that people were sat around in seiza doing za-zen related breathing exercises. At this point I had a MA background in mainly Jujitsu and Ninjutsu, and we'd never done breathing exercises of any kind, the most we'd been told is "breath in through your nose and out through your mouth", and "exhale when you strike or are thrown". We'd never even been told about breathing to or from our centre. Come to think of it, we'd never even been told about the seika tanden. Za-zen was certainly eye-opening to me (in the literal sense of increasing my awareness / zanshin, but also in the sense of "Hey, there's something more to this").
My own teacher tends to approach his Aikido in a very spiritual way, but not really talk much about that side of things, and let us reach our own conclusions / ask more / study more oursleves / etc if we decide we want to know more about it.