I'll bite, because I've been thinking about this since I read the NYT piece this morning. My info is 2nd hand, because it's based on what I've observed from my S's experience:
1. The core not only provides advantages as a curriculum, but means that all first and second year students will have some small, discussion-based classes. While the teaching, as everywhere, is mixed, those grad students involved with the core do get instruction and guidance, and there is a core office that's invested in keeping some level of quality -- unlike TAs at other universities and in other courses. My S had only grad students for all his core classes, yet his LitHum and CC were excellent, and overall the core met its goal: encouraging students to think and express themselves on a range of subjects (not just to regurgitate received wisdom.) One of the complaints in the Harvard article was that this didn't always happen there. I think my kid is getting a great education overall.
2. Faculty contact very much depends on the department. The Harvard student featured in the article majoring in German and Classics said he chose his classes carefully to maximize seminars. I suspect a student at Columbia majoring in German and Classics would also have plenty of faculty contact, while a student majoring in economics at either Harvard or Columbia would have to struggle for it, or at least strategize. What I've found with my own kid is that he's been reluctant for some reason to approach profs and has never chosen classes based on their likely size (as opposed to Culpa reviews

). Yet, when he has contacted faculty, they have been almost universally responsive, and offered him research opportunities. So my impression is that while Columbia advising stinks, many Columbia faculty are willing to give time to undergraduates who take the initiative, and there is a general faculty culture that cares about teaching. Example: I know my kid had a science lecture course in which, when the class did poorly on the first midterm, the Prof apologized to the whole class, and said he'd only give himself a "C" for how well he'd been teaching!
I really encourage students to consider departmental differences when choosing majors, and when picking classes. You can be a doctor or lawyer or investment banker with a lot of undergrad majors -- why not pick an undergrad-friendly department if given the option. Also, go to see professors when you're excited by something in class and want to follow up on an idea, not just when you want to find out what's going to be on the next exam, or complain about points that were deducted on the last one, or when you suddenly discover you need a letter of recommendation in a week. I give you this advice for free, as someone who has been a prof.
3. Star professors ARE often a mixed bag. Brinkley, I think, is supposed to give a good lecture class.