Even the useful side of math -- the stuff supposedly taught in service to the science and engineering courses -- is taught with a computer nowhere in sight. Yet people who use math in those fields, including myself, always do it with the assistance of a computer, whether it's to help avoid dumb mistakes or to automate tasks involving large amounts of numbers. And the physics problems that can really be solved with purely symbolic math are few and far between, and rather contrived. It actually leads students to believe that the formulas are false when applied in the real world.
I don't think teaching computation would harm the handful of students who develop an interest in math as a liberal art. If they want to explore, they will, either on their own or with guidance. There's no reason why they can't be introduced to abstract math, and proofs, along the way.
My kids did lots and lots of drill. But no proofs.
I struggled with math until we started to do proofs. Then it came alive for me. I loved sets. My school also used a series of textbooks in which some of the problems had no answer, and you were supposed to write "no answer." Those problems were a special rare treat that motivated me to do all of the problems.
For most kids and their parents, math functions as some sort of diligence / obedience / IQ training that they hope will get them into a better college and job before it is forgotten.