As with “elementary” you have to see this in the broader context of Quantum Mechanics, a truly weird and Byzantine theoretical architecture. Anything that isn’t pages of bra-ket notation with occasional words “naturally follows from...” is elementary. Anything that isn’t just a load of linear algebra, but includes the odd diagram is a “picture book.”

Quantum Mechanics, it’s amazing, but it’s hard to wade into.

Are there opportunities for improving its representational design?

It’s possible to do this subsets of stuff, but rewriting QM in a better way is at least as difficult as rewriting the linear algebra of hermitian operators in a way that is easily palatable.

While QM is somewhat of a new development, linear algebra has been around for nearly 150+ years in its roughly modern form (and far longer in less general forms, since around the classical Islamic empire). It is possible we now have better abstractions, but this is also generally unclear.

No. The stuff that exists works well enough, and isn't anywhere near as complicated as what it represents so it's not worth changing it