Who says they are not well organized?
Trust me, if the NCAA Eligibility Center had to make decisions prior to the start of school semesters, a huge number of kids would not be able to play as freshman.
A big part of the delay in clearing student-athletes is that many of these kids wait until it is very late to submit all of the documentation the NCAA requires. It is a shame really, because the NCAA Eligibility Center website provides dates by which each set of documents should be submitted. Sadly many kids do not even register with the NCAA Eligibility Center until late in their junior year of HS. I know of one kid who was a senior in HS and I mentioned the NCAA Eligibility Center to his father and he straight up told me that he didn't know what that was.
Now, mix in a kid who started HS in another country. The NCAA has to verify that the kid actually attended a real school with legit classes. The NCAA may have to send someone to the kid's native country to visit the school. Now mix in a kid who reclassifies. The NCAA has to verify that the kid completed certain core classes before the start of what is their "new" junior year.
It is not the most difficult thing but some students make it hard on themselves by procrastinating.
It is the student-athlete's responsibility to register, submit documents, have their HS submit transcripts and have wherever they took the ACT or SAT submit test scores to the NCAA. Get the info submitted late and the NCAA doesn't move you to the front of the line just because your sport starts practice in late September.
With regard to kids from other countries, I occasionally help out a kid who wants to go to the US for college from New Zealand, sometimes athletes.
Kids in NZ do not "graduate" from high school. There is no such thing here. Kids work toward "qualifications." A Level Two Qualification is necessary to go to college. College admissions here are pretty much open if you have a Level Two Qualification, or are 20 years old. It isn't competitive except in a few areas (there are a couple quibbles on what I've said, but it's 95% accurate).
So, when the US comes looking, the Kiwi hasn't graduated from HS. Nor does he have a GPA. They don't worry about grades here. You have to pass tests to get credit in courses, and the credit system goes "Not Achieved, Achieved, Merit, and Excellence." There is no college admissions test.
And we're not even a third world country (although you cannot get a decent bagel in the whole country)! So, it can get complicated. BTW, kids who are done with HS are not called "high school graduates," they are called "high school leavers" -- I kid you not.