I think this is in general quite insightful, but I want to pick one nit (in reaction to "A professor doesn't tell you when to..."): having advised a number of PhD students (in applied math), I think one of the most important parts of the job of an advisor is to coach. That is, help students make decisions, set timelines, figure out when to do what. Most students who get this far are capable of solving well-posed, textbook-sized technical problems. What they have trouble with is how to break a big problem into smaller ones, which of the smaller problems to solve, in what order, how to prioritize, where to compromise, etc. Good advisors help beginning students do this. (Not that I'm good at this -- I've certainly learned this the hard way over the years; just hoping the students didn't suffer too much as a result.)
My biggest problem is writing/presenting efficiently. Usually I spend several hours on a single paragraph or beamer slide. Today I spend all my time preparing my defense presentation (besides social media), but only made drafts for two slides.
And my advisor is also very busy. He says he will write the introduction for a paper and then finds no time to finish it
The making of slides (or sentences in a paper) gets easier with practice. The advisor or collaborator or you yourself being too busy is, unfortunately, not something that gets better with time. The most important thing is learning to say "no" to new responsibilities without becoming an irresponsible member of whatever organization (university, research lab, etc) you find yourself in. It's a balancing act.
