I think the real key, here, in lineage - what is the person's link back to O'Sensei. If an instructor is open about explaining that, then they are probably going represent themselves all right in general. They may or may not be good teachers, nice people, etc., but at least you'll know their background. On the other hand, if they claim an oddly high rank, or to have beaten Chuck Norris in thumb wrestling, etc, then there may be a problem.....
PS For what it's worth, here's how I would answer that question: I'm sandan, I've been training for 20 years, and I started a dojo so I could keep training. My first senior teacher was Akira Tohei Shihan. Now I am a member of the ASU and am a student of Saotome Shihan (through his senior students) and Ikeda Shihan (who promoted me to my current rank). My other teacher, and the central influence on my aikido training today, is Mary Heiny Sensei. Tohei Sensei and Saotome Sensei were direct students of O'Sensei. Ikeda Sensei is Saotome Sensei's senior student. Mary Heiny Sensei is a student of Hikitsuchi Sensei (10th Dan), who was a student of O'Sensei for decades.
I agree strongly. I have come across quite a few dojo pages where no lineage is given for the instructors, and in my opinion this makes it very hard to assess the dojo, short of actually going there in person. One example I found recently is this one, not too far from me:
They make wild statements about the present state of aikido, but there is no information that I can find on the site about where the teachers have trained before, or who awarded their grades.
Perhaps Tony Wagstaffe or Henry Ellis know Antony Day?
You will find similarly strongly-worded personal opinions on Henry Ellis's site, but then at least he does have a pedigree!
By the way mine is actually pretty simple: I started training in Oxford in the late 1970s, and for most of the time since then I have had only one teacher, Minoru Kanetsuka, who was originally a student of Gozo Shioda and then of Kazuo Chiba. My dan grades are all certified by Aikikai Hombu Dojo.