I think that there are likely some problematic aspects to this idea. One is that the general "I learned X, therefore I can learn Y" seems really strongly applicable to all STEM-types of classes, but I think it would likely falter in English, History, and related subjects to some extent - for example, in history, there's a lot of problems trying to organize how and what to learn that mean making a simple graphlike structure seems problematic. All events have preceding events and resulting events necessary to place them in their proper context. You can't really understand World War 2's causes without understand World War 1 and it's resulting effects, and you can't understand that without understanding.... etc. And while you could start all young children at "billions of years ago, the Earth ..." and progress until you are teaching them recent history at the end of highschool, that will necessarily mean they will understand all the earliest stuff at a very low level of sophistication; and you can't really teach the entire history of the world at varying levels of sophistication each year, or even every 3-4 years. And finally, there's really no way to test "Does this student understand the history of the Opium Wars?" in an automatic way that is resistant to cheating and allows retries without being heavily proctored by actual physical humans standing over watching the testing and writing very carefully and doing the marking of essays, which rather defeats the benefits of a lot of this flexibility stuff for those fields. Also, these fields have a lot of required skills that all should be advancing together - history and politics and economics and so forth inform one another a great deal, for example. I have no idea how I would apply something like this to literature and reading skills, etc. either - the dependencies for these fields are far more fluid and less obvious than "Before you can learn subtraction you must learn addition", and necessarily involve a variety of skills in. e.g. vocabulary, grammar, spelling, composition, and the like all rising in somewhat non-discrete ways. The extent to which learning how to spell "definitely" allows you to write memos is extremely non-obvious, to me at least.
Secondly, people don't necessarily know and understand things perfectly because they passed a test on it some time ago. I personally have a very strong memory, and didn't need to be retaught how to find the roots of a quadratic equation after the first time. However, I was personally taught that particular concept at least 3 times that I can recall in public schools, and even the third time I would say that more of my classmates NEEDED it than not, which is depressing but important to realize. I am not immune to it myself - I only retained permanently "how to invert a matrix and why it matters at all" on the second go-round of that concept, and I saw it at least half a dozen times. You have to try to deal with that sort of thing - perhaps retesting before each new concept? Or some time-based decaying proficiency individually for each student based on their history? Regardless, however you attack this problem it is going to be a problem for this sort of approach that ought to be considered.
However, those are sort of nitpicks, really - there are several more fundamental problems. Overall, I don't really see that this project will significantly advance the field of pedagogy relative to the resources it requires to even begin to attack it. A lot of your categories have no suggestion that I consider reasonable for even beginning to attack the roots of the problem as I see them. For example, your section on writing merely states that we should teach people to be good writers somehow, and your rationality section that we should teach them rationality somehow. I found personally that the majority of my teachers in school were not fundamentally extremely competent writers and powerful orators who were able to pass on their gifts of persuasive composition effectively, and similar limits applied for logic and rationality - there is a fundamental chicken and egg problem where you want a lot of people to learn X but there aren't any teachers who currently even know X and little reason exists to suggest that they would be particularly good at teaching X*. And there are a huge number of students who learn best from a teacher, and whom I suspect would make little real progress if left alone. Independent learning works best, so far as I know, for prodigies who are very self-motivated by long-term goals, which characterization I suspect applies to rather less than 1% of the general student body of a typical modern school - Lesswrong's audience is, I suspect self-selected for the sort of people for whom this idea might not be terrible, but there are a huge number of students who would quite simply cheat if left to their own devices with a system such as you suggest, or do minimal real learning but instead cram for tests immediately before taking the tests and then forget them. You could enforce proctoring of official tests to advance, but that seems like it would quickly become a bureaucratic nightmare and loses a lot of the flexibility and so forth you wanted in the first place.
Anecdotally: I personally went to all of my scheduled my classes through all my undergrad because I already knew by that time that I learned best by going to a class where I could listen to a presentation and ask questions, even when the class material was adapted directly from a book I already had in my possession from the first day of class and I had complete sample problems sets similar to those on the final exams as well as solutions. I rather suspect more students are like me in this regard than not, because the regular attendance rates of those classes was >90% even during extremely nasty weather, and they all had the book and problem sets too.
I think that there are likely some problematic aspects to this idea. One is that the general "I learned X, therefore I can learn Y" seems really strongly applicable to all STEM-types of classes, but I think it would likely falter in English, History, and related subjects to some extent - for example, in history, there's a lot of problems trying to organize how and what to learn that mean making a simple graphlike structure seems problematic. All events have preceding events and resulting events necessary to place them in their proper context. You can't really understand World War 2's causes without understand World War 1 and it's resulting effects, and you can't understand that without understanding.... etc. And while you could start all young children at "billions of years ago, the Earth ..." and progress until you are teaching them recent history at the end of highschool, that will necessarily mean they will understand all the earliest stuff at a very low level of sophistication; and you can't really teach the entire history of the world at varying levels of sophistication each year, or even every 3-4 years. And finally, there's really no way to test "Does this student understand the history of the Opium Wars?" in an automatic way that is resistant to cheating and allows retries without being heavily proctored by actual physical humans standing over watching the testing and writing very carefully and doing the marking of essays, which rather defeats the benefits of a lot of this flexibility stuff for those fields. Also, these fields have a lot of required skills that all should be advancing together - history and politics and economics and so forth inform one another a great deal, for example. I have no idea how I would apply something like this to literature and reading skills, etc. either - the dependencies for these fields are far more fluid and less obvious than "Before you can learn subtraction you must learn addition", and necessarily involve a variety of skills in. e.g. vocabulary, grammar, spelling, composition, and the like all rising in somewhat non-discrete ways. The extent to which learning how to spell "definitely" allows you to write memos is extremely non-obvious, to me at least.
Secondly, people don't necessarily know and understand things perfectly because they passed a test on it some time ago. I personally have a very strong memory, and didn't need to be retaught how to find the roots of a quadratic equation after the first time. However, I was personally taught that particular concept at least 3 times that I can recall in public schools, and even the third time I would say that more of my classmates NEEDED it than not, which is depressing but important to realize. I am not immune to it myself - I only retained permanently "how to invert a matrix and why it matters at all" on the second go-round of that concept, and I saw it at least half a dozen times. You have to try to deal with that sort of thing - perhaps retesting before each new concept? Or some time-based decaying proficiency individually for each student based on their history? Regardless, however you attack this problem it is going to be a problem for this sort of approach that ought to be considered.
However, those are sort of nitpicks, really - there are several more fundamental problems. Overall, I don't really see that this project will significantly advance the field of pedagogy relative to the resources it requires to even begin to attack it. A lot of your categories have no suggestion that I consider reasonable for even beginning to attack the roots of the problem as I see them. For example, your section on writing merely states that we should teach people to be good writers somehow, and your rationality section that we should teach them rationality somehow. I found personally that the majority of my teachers in school were not fundamentally extremely competent writers and powerful orators who were able to pass on their gifts of persuasive composition effectively, and similar limits applied for logic and rationality - there is a fundamental chicken and egg problem where you want a lot of people to learn X but there aren't any teachers who currently even know X and little reason exists to suggest that they would be particularly good at teaching X*. And there are a huge number of students who learn best from a teacher, and whom I suspect would make little real progress if left alone. Independent learning works best, so far as I know, for prodigies who are very self-motivated by long-term goals, which characterization I suspect applies to rather less than 1% of the general student body of a typical modern school - Lesswrong's audience is, I suspect self-selected for the sort of people for whom this idea might not be terrible, but there are a huge number of students who would quite simply cheat if left to their own devices with a system such as you suggest, or do minimal real learning but instead cram for tests immediately before taking the tests and then forget them. You could enforce proctoring of official tests to advance, but that seems like it would quickly become a bureaucratic nightmare and loses a lot of the flexibility and so forth you wanted in the first place.
Anecdotally: I personally went to all of my scheduled my classes through all my undergrad because I already knew by that time that I learned best by going to a class where I could listen to a presentation and ask questions, even when the class material was adapted directly from a book I already had in my possession from the first day of class and I had complete sample problems sets similar to those on the final exams as well as solutions. I rather suspect more students are like me in this regard than not, because the regular attendance rates of those classes was >90% even during extremely nasty weather, and they all had the book and problem sets too.