So... If gravity is quantized, then black holes must be only a low-energy approximation of whatever phenomenon is really happening there?

If there's some proof that they aren't black (AFAIK, the only thing we know empirically), I've missed it. All I see is a point that the current theory would be wrong.

Rather, the existence of black holes demonstrates that gravity is not renormalizable.

The last sentence in the paper:

"It seems that gravity is a low energy effective field theory description of something else that is not a quantum field theory."

QFTs like QED and QCD are renormalizable. This is a technique used to eliminate the infinities that arise in calculations from self-interaction. For a very long time, renormalization was viewed as hocus pocus (including by the person who discovered it). Later, mathematicians were able to provide a solid theoretical foundation for it.......but only as an effective field theory valid at particular size and energy scales.

Net net, the standard model is an approximation of something more fundamental. Gravity being nonrenormalizable shows that "something" is not a QFT.

Does it contest anything that we have observed? Because I couldn't find anything there that won't allow objects that behave the same way as the black holes we've seen, and differ only on details we can't see. (But IANAP, and could easily have missed something.)

There are tons of effective QFTs that can be used very accurately to predict experiments below a certain (high) energy threshold, but which are not renormalizable. This is well understood. It means that the theory needs to be augmented to understand the higher energies, but does not at all mean the more fundamental (augmented) theory has to be radically different, such as classical. Everyone already agrees that the non-renormalizability of naive low-energy quantum gravity means something must be added.