Is the rei necessarily spiritual at all? My interpretation is that the baseline is acknowledgment of, and respect for, what has come before, and that is how I explain shomen-rei to my class. You can add whatever sentiment you like on top of that.
I like Kanshu Sunadomari's statement that aikido practice itself can be thought of as a spiritual exercise. Again I don't believe that it is necessarily so for everyone, but I think I understand what Sunadomari Sensei means.
Spirituality varies by degree - I would say that rei certainly is - but that does not necessarily mean there is a religious connection and if there is (ie. the kamiza or seat of the Gods) that aspect can be safely ignored.
By the by - the way you describe rei is exactly how I do. Its use has the added effect to prepare yourself for what is to come. It certainly does not fall into the technical realm of Aikido.