In Cheng and Li's book, Gauge theory of Elementary Particle Physics, he essentially says that renormalization has nothing to do with infinities. Even in a totally finite theory, we would still have to renormalize physical quantities. For example, the mass $m^*$ of an electron inside the crystal is renormalized from the mass $m$ it has outside the crystal (due to the interaction inside the crystal). However, unlike relativistic QFT, both $m$ and $m^*$ are measurable and finite. Therefore, the correction $\delta m=m-m^*$ should also be finite.
How does one calculate this correction $\delta m$? If one uses quantum field theory, he finds that the correction to electron mass is logarithmically divergent.