I think it didn’t get included into the five senses for the same reason you call it “the sense of balance” in your post: it is usually thought to be a sensation, not a sense — meaning that it tells you not about what’s outside (like sight, hearing, or smell), but about the state or condition you’re in. Same goes for pain, for example: it gives the information about your organism, not about the outside world, hence not a sense in “the five senses” sense.

The sense of balance you’re talking about is in fact undisputedly a sense — a sense of gravity — but I guess since the vast majority of people have no experience of being in situations where the sense of gravity is useful for getting more information than the balance, and the balance is an “internal” thing, we have yet to see it “promoted” to senses — yes, I do hope the general public will get out of the gravity well one day…