No disagreement there, and especially the way it's normally taught now, which can feel more like a course on special-case methods for performing symbolic integration. There is some understanding you gain by knowing how to work symbolic integrals (the chain rule, integration by parts, etc., etc.), but I wouldn't put it near the top of the list of things all students must know. The actual integration can be done fine by Mathematica or Maple; what's important is what you do with that.