Why does the ouput of an NC0 circuit depend on only a constant number of input bits?
An $\mathsf{NC}^0$ circuit with single output of depth $\ell$ has $O(2^\ell)$ nodes, and in particular can depend on at most $O(2^\ell)$ many input bits.

$\mathsf{NC}^0$ circuits become more interesting when we allow them to have many outputs. Amazingly, there are some things you can actually do with some circuits. Check out the classic paper Crypography in $\mathsf{NC}^0$ by Applebaum, Ishai, and Kushilevitz.