When I see 15/14 I know he is tight, but he's also being agro with basically any hand he plays. That range includes a lot more than AK, AA and KK. In other words, we're still way ahead of most of it, so I think jamming is ok.
I don't think there's anything wrong with shoving, but I do agree with dbitel in that smooth calling will give us a slightly better EV play, for reasons already mentioned.
I understand slow playing may extract more value, but I'm also happy taking it down with all-in PF. Obv, you can play it either way. I think for a less skilled player such as myself, the easier all-in PF is correct. For a really good post-flop player, calling is probably best. But I still think it's close.
I don't think you need to be a 'really good' postflop player - you pretty much shove when he c-bets a favourable flop, check/fold an unfavourable one. The only spot it would get tricky is on an Axx or Kxx flop, and having to decide whether he's got it or not.
I do see your point, though - I'm pretty much a terrible postflop player vs my preflop play, and although I agree smooth calling can extract a little more, I probably shove in the moment.