"Basic (and useful) mathematics is about calculations and higher (pure) mathematics is about proof."

One reason I think this is harmful is that there is no sharp line between calculations and proofs. Very often a certain calculation is essentially the proof except for a few logical connectives. Conversely, in formal logic, one can create a "calculus" that makes proofs appear to be calculations.

Another reason is that it leads students (and more importantly teachers!) to think that a drastic change of mindset is required to learn higher mathematics.

It is indeed true that analysis is quite different from calculus even though there is strong linkage. However, the former also leads to better techniques to calculate things. Putting too much emphasis on the "proof" aspect of analysis tends to put off a lot students who enjoyed playing with polynomials, trigonometry and calculus. Conversely, many students who like to work with proofs are encouraged to believe that what they are doing is somehow "superior" (higher) to "mere" calculus; they then do not do enough computational drills which ill-serves them if they actually take up mathematics!