I would agree, it makes them anything but elementary. I am honestly not even sure if there is a finite constructible basis of the functions that can express any solution of single-variable integer polynomials.

And for multivariate polynomials, the roots are uncomputable due to MRDP theorem.

It is not known, and the model problem for this is Hilbert's 13th [1].

Nonetheless, "elementary function" is a technical term dating back to the 19th century; it's very much not a general adjective whose synonym is "basic".

[1] https://en.wikipedia.org/wiki/Hilbert%27s_thirteenth_problem