What adds to the practicality of pushing hands are Taijiquan's unique principles of softness and coordination instead of hardness and brute strength. This makes push hands a gateway for anyone, old, young, big, small, male, female, etc., to be able to acquire an efficient technique for self-defence.
As I mentioned above, it is also the most important "test" for a Taijiquan teacher. If the teacher doesn't know pushing hands, they have no business teaching. If your movement doesn't have the leverage to work in order to move another's centre, or to protect your own, how can a student know what you are showing them is correct? A teacher should be able to demonstrate consistently and conclusively that their execution of the techniques in the forms are accurate as advertised. A student should be able to directly see and feel what is being taught, otherwise how can they know they aren't being fed a line of smoke and mirrors?
Healing was mentioned by another poster above, without pushing hands, without knowing what realistically affects and doesn't affect joints and muscles, how can you realign them? There are other disciplines, sure, but a massage therapist can't hold a candle to a traditional Taijiquan teacher's tui na abilities.
I was fortunate to have spent time and to have had classes with Wu Daxin laoshi and his disciples (esp. Dr. Wu, no relation) on the theories and applications of Taijiquan related tui na, acupuncture, herbs, etc., as well as the positionings relative to others we in the west might call "psychology."
I know I'm downloading about this issue on the Yang family discussion board a lot, but the question is one that comes up frequently in my teaching career and I have a lot of answers ready. I hope it is interesting, at least.