Quantum field theory is a very delicate thing. The history of its development is also intricately intertwined with perturbation theory because for most of its development, there were very few techniques for answering questions non-perturbatively. As a result, large swaths of the language are still tied to perturbation theory.
So while the speaker you listened to at your university may have been talking about finding a QFT which is renormalizable in the power-counting sense, which is the sense required by perturbation theory with a finite number of counter terms (it's interesting to note, by the way, that renormalization works even for non-renormalizable theories, it's just a matter of needing infinitely many renormalized couplings). But it's also possible that they were using the term renormalizable in the sense of the theory being UV complete.
That is, the requirement that the theory flows in the UV to a sensible theory (not necessarily a fixed point) under the renormalization group flow. You will note that this point of view is completely independent of any statements about infinities which may or may not appear in specific diagrams.
I will also mention off-hand that renormalization is inescapable in quantum field theory, even non-perturbatively. For example, you can prove, using only completely non-perturbative methods, that the so-called wavefunction renormalization (the rescaling of our fields) must happen in any interacting theory. With this in mind, we really can think about the RG flow as a non-perturbative concept.