Just to throw in a metaphor, rather that Newtionian and Quantum mechanics, perhaps the schools of thought are like the differing formulations of classical mechanics like Newton Vs Hamiltonian Vs Lagrangian formulations.
There is no upper or lower bound on the minimum length of a proof when you are free to choose the language in which it is expressed. This means that there almost certainly are things can only be communicated in one language but not another, even if the languages are equivalent in some vague limiting sense.
So as a practical matter, you have to work in a better language, because the difference between spending 30 years to learn something and 30,000 years is quite significant.
This other school-of-thought business (Bayesians vs. frequentists, pilot waves vs many worlds vs Copenhagen, everything like them) means choosing between different sets of words to describe the same thing. You are left with one of two cases: either they can be shown equivalent (in which case people are prone to keep on arguing over which one is better), or they are different (in which case one is wrong.)