Quantum mechanics solved a bunch of real world problems that were relevant at the time. As one example, people were trying to predict heat capacities of gases

Quantum mechanics solved a bunch of real world problems that were relevant at the time. As one example, people were trying to predict heat capacities of gases (for things like chemical engineering industrial processes) and the theory they had, based on equipartition, produced results that didn't match experiment at room temperature. And we're not talking a few percent mismatch; we're talking significant differences. Special relativity likewise addressed specific experimental problems people were having. General relativity you may have more of a case for.