There are certainly situations where it suffices to consider the diagram of finite groups, e.g., the image of any continuous map from a profinite group to $GL_n(\mathbb{C})$ is finite. If you are only concerned with such families of representations, you don't have to worry about profinite groups.
On the other hand, there are situations where we want to look at continuous representations (e.g., of an profinite Galois group) on vector spaces with finer topology (e.g., $\ell$-adic or $p$-adic groups) and these often have infinite image. In principle, it is still possible to view these as compatible systems of representations on groups like $GL_n(\mathbb{Z}/\ell^k\mathbb{Z})$ for $k \geq 1$, but to me, it seems easier to consider a single continuous map and let the topology do its job.
Similarly, we may want to look at Galois representations that are slightly discontinuous (e.g., where the image of some Frobenius has infinite order) - one typically fixes this discontinuity by choosing a subgroup of the Galois group where Frobenius lifts don't generate compact groups. If we try to view this as a compatible system of finite Galois representations, it seems to be somewhat cumbersome.
