That theorem is not formulated about "elementary functions".

It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals".

While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention.

The usual convention is to include all algebraic functions and a few selected transcendental functions.

In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations.

Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals.

For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation.

So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.