Maybe educational students should only require what's actually practical. That can include things that can be solved by computers if they teach ideas that have practical value.
Yes, but the logic goes the other way: if they are unable to use the technique manually, then one may conclude they do not understand the idea well enough to use it, whether or not Mathematica is available. The two, manual application and understanding/intuition, go hand in hand and can't be separated. My understanding is that this has even been studied more formally, there is a paper I remember reading by Kahneman and somebody else  about the development of intuition and the tension between heuristics-and-biases and naturalistic decision making. In short, don't trust people so much, if they say they understand something but are unable to actually do it, be a bit more skeptical.
 Found it: https://www.ncbi.nlm.nih.gov/pubmed/19739881 On second thought, it's probably not as relevant as I remembered it.