Tensor algebras are the most general algebras you can get (categorically inclined people would call them a “free object” in a suitable category), but because of this they are almost never useful on their own. We pretty much always study specialisations of tensor algebras, of which GA (or more generally Clifford algebras) are an example.

There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no relations. It is only when passing to more specific algebras (symmetric, exterior, GA, ...) that the subject really becomes interesting.