Sure but you make an arbitrary choice here, you choose to use Cartesian coordinates. Lagrangian and Hamiltonian mechanics make it more clear that this is more or less an arbitrary choice and that there are other sets of generalized coordinates to describe the problem, something that is in some sense less obvious in Newtonian mechanics because things usually become pretty hard to deal with.
Yes, every choice of coordinates is capable of describing the physics of a system. Yes, you could certainly argue that Cartesian coordinates are a natural choice or in some sense special, derivatives are especially simple and whatnot. But I still think that something is lost when one disregards rotations and angular momentum, they seem to capture an important aspect of the structure of space, its isotropy.
Then again every equivalent description should capture the same things, just maybe not in an obvious way. And then again the existence of spin angular momentum hints at the fact that there is really more to it. As I said, I am really not sure. I used to think of rotations as emergent from translations and I kind of changed my mind but I am probably still on the edge.