"I plan to remain hard-headed about this until I am convinced that abandoning the rote sorts of exercises done in, say, a linear algebra class (which can also be done on a calculator) does not hinder our ability to form intuition about how to do proofs, etc"

It is positive statements that require proof, not negative ones. If you believe that the introduction of calculators, google maps, etc., has negatively impacted number sense and human spacial reasoning, it is on you to prove it.