To me, a requirement of what would qualify as a spiritual practice would be the element of personal transformation. O-Sensei clearly envisioned that Aikido be a means of first personal transformation and then by extension, a means to transform the world. But I think it is not clear at all whether succeeded in developing a training method that is going to accomplish this on any wide scale.
For example, Zen training has for a couple thousand years been a fairly systematic practice (varying from sect to sect in emphasis) that leads to an experience of satori or enlightenment. Regardless of what style of Zen you do, Soto, Rinzai, eclectic, there is a system for evaluating the level of insight on the part of the student and a system of transmission from teacher to student.
Aikido on the other hand is completely lacking in any actual agreement about what spiritual practice is. For many, this simply means that there is an ethical dimension to the practice. It fosters a caretaking attitude towards all life and contains a non-violent philosophy that is reflected in the manner in which we execute technique.
It is clear that O-Sensei had a mystical vision of what Aikido was. For him, the execution of each Aikido technique was a symbolic representation on an energetic level of the reality of the universe. Every detail of a technique contained the symbolism that allowed this expression. Yet very few people are doing that in their training.
There are a variety of lessons to be learned from training in Aikido on a personal level. But I find that often the descriptions many practitioners give when describing the "spiritual" benefits of the training they have derived are pretty much the same descriptions that people who have seriously devoted themselves to any other endeavor give. I hear exactly the same descriptions from serious athletes when they talk about what training in their particular sport (or dancers or musicians) has done for them.
Any activity that requires being "present" will cause some type of personal transformation. Our society encourages us to be asleep so by contrast any activity that doesn't will be beneficial. Our society encourages us to look at everything from a labor saving point of view so anything that involves purposely looking for difficulty will be transformational.
What people mean when they refer to Aikido as spiritual practice seems to range from simple pleasant mutual admiration societies that many dojos are (the focus is on not doing anything that threatens a members self esteem) to beat the body strengthen the spirit approaches to practice (which mostly harden the "will" of the students without necessarily doing anything to really transform them)to those who are on some level trying to understand their practice in some way similar to the way in which O-Sensei viewed it.
I think that Aikido has the potential to be a very deep practice that could afford deep personal transformation. But I do not think that there is anything within what is generally called Aikido that we can say we all hold in common as to what the method is or even what the spiritual goal of this practice is. It is completely different from teacher to teacher and student to student. It will be interesting to see what we all come up with over time; maybe some will really come up with something deep.