Firstly, renormalisation isn't really inherently to do with the UV divergences you get: it's more to do with the idea that interactions with other fields change a particle's energy from its inherent 'bare' mass, so the measured mass isn't the same as the value that appears in the quadratic part of the action. You play a renormalisation game in perturbation theory to make the parameters that appear in predictions match with simple measurable quantities, not Lagrangian parameters. You would want to 'renormalise' the parameters like this even if the radiative corrections were finite.
Having said this, there are (at least) two reasons why renormalisable theories are important:
Firstly, you need only fix a finite number of parameters in this way, and thereafter everything is computable. For nonrenormalisable theories, you lose a lot of predictive power because infinitely many adjustments are required. This was historically why renormalisation was important: it showed that you could consistently follow the rules in QED, say, and get answers at every order in perturbation theory.
Secondly (the more modern perspective on renormalisation, associated with the name of Wilson), at low energies, every theory will look like a renormalisable one, and you don't care too much what happens at high energies. The infinities that appear in loop calculations are a manifestation of some unknown stuff going on at very high energies, and the fact that you can get rid of them with renormalisation is related to the fact that the detailed form of the high energy theory is unimportant, showing up only as the precise values of some finite number of parameters at low energies.
