Why search for a renormalizable theory of quantum gravity?
There are two perspectives (or rather two ways of saying the same thing), according to whom you ask, they might prefer one or the other:
Non-renormalizable theories need an infinite number of counterterms. While that is perturbatively ok, it leads to a theory that is ultimately non predictive as all these counterterms need to be fixed by infinitely many experiments.
Non-renormalizable theories are typically the sign of having an effective field theory. Namely, they are a result of a low energy expansion of an underlying UV complete theory. Our goal is to figure out the UV theory.
The expectation that the UV theory will be a QFT is probably not very popular as of now. We have to either confide in string theory or come up with a new paradigm entirely. Don't get me wrong: there are people working on the assumption that there is an interacting UV fixed point that leads to gravity in the IR, but I am not expert enough on that to comment.
About 1. : Why do you need infinite counterterms? Because that's what non-renormalizable means and if you don't renormalize a theory, the stuff you compute gives $\infty$.
About 2. : What do you mean "typically," what are other examples? This is by now almost a sacred paradigm of phenomenology. It comes from looking in hindsight at all the developments of early QFT that led to the Standard Model. At the time we had the Fermi theory, which was non-renormalizable, and later we figured out that there were heavier particles modifying the theory in the UV.
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.
There are various views on this topic as far as I understand but I would put forth here my understanding.
The basic reason as to why we would expect a quantum theory of gravity or any theory of everything to be renormalizable nonetheless is not because we expect some other theory behind the curtain from which we want to shield ourselves but because we expect our theory (which might be complete) to also be able to make useful predictions at small energies/large distances without having to worry about the high energies/small distances structure of the theory. It is the same reason why naked singularities are not desirable even in a quantum theory of gravity, not because we expect to not be able to deal with the singularity in our quantum theory but because we want general relativity to be a useful theory at low energies/large distances which it can't be if naked singularities exist which would require even low energy computations to take into account the details of the high energy theory. See this nice answer to an old question of mine.