I see quite a bit of spirituality at my dojo, but I also train at a Shinto shrine and my teacher is a priest. I think this tends to attract people who are looking for a spiritual practice or who otherwise seek that kind of environment, but not everyone fits that description. I would describe many of my fellow students as being pretty spiritual, but my sense of the word is based on having a sense of deep awe for nature and an effort to connect with that awe throught one's activities...so it's pretty open-ended.
Speaking from my own sense of the word, and from what I understand of Shinto, any activity can be "spiritual," but we do have focused meditation times. Saturdays, for example, people can do misogi no gyo, daily Shinto ceremony (chyohai), and formal Chinkon no gyo before Aikido practice. After each practice session that sensei leads there is an abreviated Chinkon no gyo.
I started Aikido because I wanted to learn something meditative that also included some kind of "martial" training. So per my own sense of spirituality, yes I joined an Aikidojo for the spiritual practice and continue now with a similar sense of purpose. My spirituality is based very much on the concept of self-improvement and being the best person I can be and the concepts of gyo/discipline and "self victory" I see in Aikido are big parts of that for me.
I am not familiar with your dojo, but it sounds as a good and interesting place to practice Aikido. The study of Shinto and practicing Shinto does in my experience enhance a deeper understanding of Aikido and the teachings of the founder of Aikido. It also brings changes to the body and mind that helps in daily life and in practising Aikido. A sense of deep awe for nature becomes part of that. Just as a feeling of connection with the community. There are not many dojo in the world where one can practice Aikido and Shinto - you are very fortunate!