Mathematicians are afraid of higher order tensors because they are unruly monsters.

There's a whole workshop of useful matrix tools. Decompositions, spectral theory, etc. These tools really break down when you generalize them to k-tensors. Even basic concepts like rank become sticky. (Iirc, the set of 3-tensors of tensor rank ≤k is not even topologically closed in general. Terrifying.) If you hand me some random 5-tensor, it's quite difficult to begin to understand it without somehow turning it into a matrix first by flattening or slicing or whatever.

Don't get me wrong. People work with these things. They do their best. But in general, mathematicians are afraid of higher order tensors. You should be too.