One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if such a curve is defined over a subfield of the complex numbers, then the rank of the Tate module is equal to the rank of the classical degree 1 cohomology of the complexified curve. We now know that the Tate module is naturally identified with étale cohomology in degree 1.

One might then reasonably hope for a similar relationship between the higher degree classical cohomology of higher dimensional varieties and certain $\ell$-adic Galois representations naturally attached to the variety.

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if such a curve is defined over a subfield of the complex numbers, then the rank of the Tate module is equal to the rank of the classical degree 1 cohomology of the complexified curve. We now know that the Tate module is naturally dual to étale cohomology in degree 1.

One might then reasonably hope for a similar relationship between the higher degree classical cohomology of higher dimensional varieties and certain $\ell$-adic Galois representations naturally attached to the variety.