The usual proof of Picard's theorem along these lines used another modular function which is called $\lambda$ and which is related to $J$ by $$J=\frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}.$$ This function $\lambda$ is simpler because it performs a $universal$ covering of $C\backslash\{0,1\}$ by the upper half-plane. See, for example, Ahlfors, Complex Analysis, pp. 306-308.

The interesting question whether one can prove Picard's theorem without the use of modular functions and coverings arose immediately after publication of Picard's proof, since mathematicians were puzzled by this proof, and found it unnatural and mysterious. Nowadays very many different proofs are available, for a brief survey of them see:

Theorems with many distinct proofs

Some of them use no coverings at all, and no monodromy; others use a much simpler universal covering $\exp: C\to C\backslash\{0,1\}$.

The usual proof of Picard's theorem along these lines used another modular function which is called $\lambda$ and which is related to $J$ by $$J=\frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}.$$ This function $\lambda$ is simpler because it performs a $universal$ covering of $C\backslash\{0,1\}$ by the upper half-plane. See, for example, Ahlfors, Complex Analysis, pp. 306-308.

The interesting question whether one can prove Picard's theorem without the use of modular functions and coverings arose immediately after publication of Picard's proof, since mathematicians were puzzled by this proof, and found it unnatural and mysterious. Nowadays very many different proofs are available, for a brief survey of them see:

Theorems with many distinct proofs

Most of them use no coverings at all, and no monodromy; others use a much simpler universal covering $\exp: C\to C\backslash\{0,1\}$.