That second point is spot on. The one thing I remember vividly - and that still helps me most - from all those years of classical piano training is the ear training. I don't have perfect pitch, but I remember A440 vividly (thanks to my electronic metronome) and the training left me with a strong sense relative pitch that helps me figure out pretty much any melody and chord progression relatively quickly.
I really wish I had studied piano for that reason. I put many years into flute and almost went to conservatory for it, so I can play pretty much any melody by ear after hearing it, but I have zero ability to do the same for harmonies/chord progressions. I've tried to learn some piano (and like the parent said, was forced to as part of music curriculum at college) but it's incredibly difficult to learn a new instrument to a satisfying level as an adult.
Can't agree more with that last sentence. I picked up guitar around 16yo, bass a couple of years later, and shortly tried my hand at drums. I'm now more than twice the age and I am indeed deeply unsatisfied with my technical ability at either.
