It wasn't wrong. It was proved for uniform materials. This paper extends it to non-uniform materials, with an additional condition.

In what way did Schauder's work turn out to be false? It simply doesn't apply to the situations discussed here.

There was this sentence in the article: "...he realized that nonuniformly elliptic PDEs that seem well behaved can have irregular solutions even when they satisfy the condition Schauder had identified"

Yes, nonuniformly elliptic PDEs. Schauder's theorem applies to uniformly elliptic PDEs.