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It appears as though we want to preserve, in a theory of quantum gravity, the empirical data that GR has led to (and most of its structure, the specifics depending on the theory of QG, e.g., we might rid of background independence). Given that much of that (such as the late merger for a binary black hole coalescence) is given by numerical relativity and other frameworks, will such frameworks be used for probing or constructing quantum gravity?

I do not really understand your question. Numerical relativity is only "solving Einstein equations in a computer". Imagine that we knew only classical Maxwell equations and now asked an analogous question about the meaning of numerical solutions of Maxwell equations to quantum electrodynamics. The answer would be that the numerical solutions are "kind of, I guess" useful for QED, but not really.

to claim a new theory as physics, after all the maths checks, one must show that the previous theory is an approximation of the new one and possibly that an old puzzle is now solved. Or better, he may predict a new effect. If experiments confirm it, the new theory has a great probability to become a discovery.

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