I went to law school in Australia, not the US, but I also had this experience --- that courses on contract law never involved reading any actual legal contracts, and that this latter exercise was almost completely absent from the whole program. Across my LLB and LLM, the only time we ever did anything like this was in a legal skills workshop in LLM (required for registration as a solicitor) where the teacher went through an actual legal contract and the class discussed the drafting of each of the sections, and their good and bad points. This exercise was probably about one hour over the course of about five years in law school.
As to why this is done, I can only assume that legal academics believe that it is more valuable to focus time on discrete legal issues, and the theory pertaining to them, rather than spending substantial amounts of time holistically analysing an actual legal contract. Furthermore, I think the law schools regard this kind of exercise as something you learn on-the-job in a legal firm, rather than in university programs. There is some degree of demarcation between legal theory and ideas taught in university (which is more abstract), and the on-the-job knowledge you get from supervising solicitors when starting in a legal firm (which is more concrete).
To give some historical background to this, it used to be the case that law schools did not exist at all. You became a lawyer by doing an apprenticeship under an existing solicitor, just as you would have to become a blacksmith, etc. Throughout the middle ages the legal profession was conducted on an apprenticeship model, so students learned skills by performing or observing tasks from an instructing solicitor. The size of the profession was small, and limited to upper-class males, so you might have had a solicitor who agrees to take on one or two apprentices who do clerical work for him while observing his practice. In the 1200s an institution called the "Inns of Court" was established in London to teach young upper-class men legal skills and the broader social graces and skills of becoming a "gentleman". Even then, formal legal training was light, and most of the learning was done on-the-job by apprenticeship under a practicing lawyer. (Incidentally, this institution is still around, and is now a professional association for barristers.) Students were also encouraged to learn by directly attending and observing court proceedings. In more recent times, when the universities began teaching law, they had to demonstrate some value-added in conducting courses through their own institutions; they did this by taking the view that they would teach the kinds of higher-level abstract principles of law, while leaving other skills to remain within the apprenticeship. Gradually the apprenticeship model has waned and disappeared, but there is still some degree to which the universities consider early career at a law firm to be a supplement for this.
I partly agree with the Sedburry quote --- it is descriptively accurate about what happens at law school. I disagree with his assumption that academics assume that concrete knowledge of how to draft good contracts follows automatically. To the contrary, I don't think that legal academics think that the ability to draft contracts automatically follows from conceptual understanding, but they do seem to regard this as a skill to be learned during one's early career in a law firm, rather than something to devote time to at university. Personally, I think that is a mistake, and that it would be beneficial for the degree to include many more exercises involving full reading/critique of a whole legal contract, even if this crowds out some of the theory knowledge in the curriculum.
Finally, I will just note that you are probably not alone in viewing this as a deficiency in the law school curriculum. This is a situation where you could take the initiative and start your own study group, with the goal of reading and analysing actual commercial contracts. You could treat this like a book-club --- each week you all get a legal contract to read and discuss/critique for the next week. If you can find ten or more other students who are keen on this, you could probably go to some of your lecturers and see if any of them are interested in helping you (e.g., attending a one-hour study session each week to help with discussion). Some academics won't have the time for this, but if you've already done the leg-work of organising the group yourself (and if you meet in a room very near to their office!), you will probably be able to find a willing academic.