Edit: There's a big danger in accepting theories because they have simplicity or beauty. That's how we ended up with String Theory and Fundamental Theoretical Physics has basically stalled since the '50s because of it.
As a historical precedent consider Newtonian vs. Einsteinian mechanics. Newtonian mechanics works fine at familiar scales of time, space, force, power, mass, etc. Einsteinian mechanics also works fine at those scales. If that's all the data you have, they are equivalent theories. Yet Einsteinian mechanics also predicts things far beyond that scale correctly while Newtonian mechanics does not.
I still think it's an interesting thing to attempt and may suggest new experimental predictions. What if an alternative model makes a novel prediction that is testable?
3D compressible N.-S. where error is measured via some functional of interest after direct numerical simulation of the discovered equations would be neat to see. I am curious which conservation laws it could figure out.