As an aside to my earlier direct reply, these are foundations-of-physics issues. Most working physicists don't really care what are in the foundations, to the great annoyance of many philosophers. (Tim Maudlin comes instantly to mind.)
But annoyed philosophers don't change the results from numerical relativity.

In https://arxiv.org/abs/gr-qc/0202079v1 Rovelli wrote something striking that I'm glad this discussion took me to:

"The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning. Seems to me that it is better not to found hamiltonian mechanics on a notion devoid of physical significance."

Sure, nobody should disagree that such a surface is unphysical. However, 3+1 decompositions are at the bottom of successful results -- in particular you will be hard-pressed to find numerical relativity projects that don't split spacetime into spaces organized by some arbitrary time coordinate, where each space is a large enough simultaneity surface to trigger his fundamental objection. Yet we have good matches to real astrophysical results, for example Ransom, Archibald et. al, https://arxiv.org/abs/1401.0535 ( http://www.astron.nl/~archibald/video.html )

Few disagree that 3+1 decompositions pose difficulties; Alcubierre wrote a book cataloguing a bunch of them ( https://www.researchgate.net/publication/228588827_Introduct... )

And of course, quantizing the Hamiltonian formulation of General Relativity gives us Canonical Quantum Gravity (CQG), which exposed the non-renormalizability (by power set counting, anyway) of gravity. That's the starting point for Rovelli's objections. However, CQG is a perfectly fine effective theory ( http://www.preposterousuniverse.com/blog/2013/06/20/how-quan... and http://backreaction.blogspot.com/2014/11/the-paradigm-shift-... form a good one-two punch on this topic).

A complete theory from which CQG emerges in weak gravity is a good goal, and Rovelli is pursuing it, while trying to keep the good bits of modern physics (in particular that we've made practically all laws of physics generally covariant or at least relativistic, and that this is not just useful, but reflects something real about the universe).

More power to him. But his success in his project has practically no impact on the success of modern uses of General Relativity (in various formulations), or its complete accord with all available evidence accumulated so far. Seriously, there is no counter-evidence. It is mathematically complete. The only thing left for it is to study the mechanisms that generate the metric and the microscopic details of sources. (Maybe we can combine the two to study what metric large quantum systems (~ milligram rest masses) actually source when prepared in superpositions of position, for example. We think there's some problem of some sort there because semiclassical gravity -- relativistic quantum field theory on a "background" of standard General Relativity -- predicts something nonsensical; maybe Rovelli's work in the foundations will find a way to make a reliable non-nonsensical prediction there by the time we can prepare such a large quantum system. FWIW, progress on that front is being made: http://sciencenordic.com/can-large-objects-exist-quantum-sta... ).