For what its worth,
When I lived in Japan a few years ago the first class I trained at Seibukan they asked me who my teacher was and I replied that it was Saotome sensei, everyone said oh you mean Yamaguchi sensei's student. I agreed then added that he was an uchi deshi as well but after several minutes of puzzled looks everyone said "Ah yes Yamaguchi sensei's student who moved to the US, very good aikido". I like to tell this story to my friend's in the US not to disrespect Saotome Sensei who I still think of as my teacher and an uchi deshi but to put in perspective that what we have been told in the US and abroad is not the "whole story" or not the "story" in Japan. I also think that other people may have similar stories about the other "uchi deshi" that were sent abroad.
I would also like to point out something that I think is very important from the youtube clip of Kazuaki Tanahashi. He states that Osensei still teaches him things today and that he learns things from Osensei all the time. This made me think that maybe the problem with all this "so and so was a good/bad teacher" is not really based on anything they were teaching but what the students themselves were learning and whether or not they learned it right at that moment or ten years later. So from my perspective the discussion as to the funny math with Saotome Sensei is silly because I am sure that Saotome Sensei still remembers things that Osensei said and did even after he had passed on. Another example is Chiba Sensei saying that it took some years for him to understand the koan that Osensei gave to him, and it wasn't until years after Osensei had died that he finally got it. My point is that people don't simply learn from things that are happening at the moment. I realize that the only measure of how well a teacher taught is by looking at the students but in my mind it is 50/50 teacher and student, and it may not be an instantaneous learning experience but one that takes years and years. Many people have pointed out that responsibility lies not only with the teacher but also with the student.
One last thing that got me thinking was the discussion of how much Osensei traveled and how could anybody really be his "student" or have really learned from him without training on a day to day basis with him. Well for most of us in the US and Europe I think the same could be said. How often do we actually get to train with the shihan that we call our "sensei" or who leads our organization. There are a lucky few who are/were able to train with these teachers either by being lucky enough to train at the teacher's dojo in a large metropolitan area or by following them on the seminar circuit. However because these teachers are highly in demand and are making a living by teaching seminars even the members of their home dojo don't get to see them all that much. Look at any shihan teaching in the US and their schedule, most of them are teaching 30-40 weekends out of the year, with travel time they may only teach one or two classes a week at their own dojo. Some people only get to see "their" shihan/sensei once or twice a year and plenty of them still claim lineage on their websites and in discussions. How is this any different than Osensei who traveled around visiting friends and students teaching here and there when he could. Also how many of students actually show up to every class all the time so even if you "sensei" is in town this weekend maybe you have to work or you have to take care of the kids etc... Which brings me back to the point that it is not just the teacher who bears responsibility.
I also liked this
As Marc Abrams suggested, it's less imperative to discuss lineages than it is to concentrate on what the Founder was really doing and work to replicate it.