The classification of spherical space forms in any dimension is covered in Wolf's *Spaces of constant curvature*, which I think qualifies as "modern". Having said that, there are discrepancies between different editions, especially for the eight-dimensional space forms.

The three-dimensional case is much simpler, of course, and does not use the full machinery in Wolf's book. As stated in the question, it boils down to classifying the finite subgroups of $\mathrm{SO}(4)$ which act freely on $S^3$. As a preliminary step, one classifies the finite subgroups of the double cover
$$\mathrm{Spin}(4) \cong \mathrm{Sp}(1) \times \mathrm{Sp}(1),$$
where $\mathrm{Sp}(1)$ is the group of unit-norm quaternions.

The action of $(a,b) \in \mathrm{Sp}(1) \times \mathrm{Sp}(1)$ on a unit-norm quaternion $u \in S^3 \subset \mathbb{H}$ is given by $a u \overline{b}$, whence the action has kernel the order-2 subgroup generated by $(-1,-1)$.

The classification of subgroups of $\mathrm{Sp}(1) \times \mathrm{Sp}(1)$ is an application of Goursat's lemma. The details are in Conway and Smith's *On quaternions and octonions: their geometry, arithmetic, and symmetry*, Section III.4.

**Edit**: As pointed out by Ryan below, I've answered the wrong question. That topology and geometry determine each other for three-dimensional space forms is well-known in cosmology as I tried to expain in my answer to this question. The paper *Topological lensing in spherical spaces*, coauthored by Jeffrey Weeks, has a nice treatment and points to two papers by Threlfall and Seifert of the early 1930s *Topologische Untersuchung der Diskontinuitätsbereiche endlicher
Bewegungsgruppen des dreidimensionalen sphärischen Raumes*, which supposedly contain this result. This, of course, may not be "modern".