I don't think this is the right idea. A new theory can invalidate core assumptions of a successful old theory, revealing it to be a coincidence that it happened to work mathematically in some regimes, it's not always a case that the old theory is simply an approximation of the new one.

For examples of the "good" kind, Newton's laws of motion are indeed just an approximation of special relativity. The Schrodinger equation is just an approximation of QFT.

In contrast, GR came in and showed that Newton's law of universal attraction is completely wrong, at the fundamental level. Sure, it predicts certain phenomena correctly, but so did the epicycles that it replaced. Similarly, QM/QFT showed that Newton's laws of motion are also completely wrong, that objects (or at least particles) don't even move according to some laws of motion, they are only described by a probability wave that moves according to some laws, and between two interactions they have no definite state and are not even localized.

And of course, GR and QFT disagree on these parts - GR generally agrees with Newton's laws of motion (with the SR corrections), and QFT generally agrees with Newtoninan gravity. But you can't use GR's gravity with QFT's laws of motion, so we know one or both are broken. A new theory is very likely to also "overrule" one or both of them and show that they are not just an approximation, but completely wrong, working only "by accident" on the scenarios where they have been tested. Especially if the new theory requires there to exist fixed space distances that all relativistic observers agree on.