You can take the profinite completion $\widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$ of $\mathbb{Z}$, then open subgroups of $G( \widehat{\mathbb{Z}})$ correspond to congruence subgroups in $G(\mathbb{Z})$.

This is completely standard, when you want to translate classical automorphic stuff to adelic automorphic stuff, and you are right that this makes everything computational more pleasant.