For your example, to make the list clearer in writing I would usually do one of the two.

I like Jazz, Rock and Roll, and Classical.

Or

I like Jazz, Rock & Roll and Classical.

(Or the other way around if a list item includes "and" specifically)

I often write in a way that flows better when talking, so I usually try to find ways like the above to make that kind of flow look more understandable in writing too. I don't think your and-less version flows well, in my way of speaking at least. But I also wouldn't tell you to stop doing that, because I can understand it, and that matters most in reality.

To me, it sounds like an incomplete list. With the 'and X', I know that X is the last thing in that set. I guess that only really matters in cases where you want to absolutely show that a set only contains certain members.

An “and” before the last item tells you it’s an inclusive list. An “or” before the last item tells you it’s a pick one. A “nor” tells you it wasn’t any of those. It’s word to wait to the last item to know what the list was, but English is a screwed up language.

You call yourself a grammar nazi, and don't use the oxford comma?