[pzhon]

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Why don't mathematicians consider the Banach–Tarski paradox a contradiction that disproves the validity of the axiom of choice?

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We don't because it is not a contradiction. A contradiction is when you can prove both A and ~A. The result mentioned in the Banach-Tarski paradox is a surprise, not a contradiction.

Incidentally, you don't need to assume the axiom of choice to use nonexistence results proven using the axiom of choice. These rely on the consistency of the axiom of choice (proven by Godel and Cohen), not the axiom of choice itself; the existence of something that can't exist given the axiom of choice would show that the axiom of choice is false.

For example, you can use the axiom of choice to construct some functions to solve Hilbert's third problem, to determine whether the regular tetrahedron can be decomposed into polyhedra and reassembled into a cube. You can construct an invariant using the axiom of choice which is different for the regular tetrahedron and the cube. Because this is a nonexistence result, the usage of the axiom of choice is removable, and the result does not actually depend on the axiom of choice.