Questions of lineage are (at their best) factual questions, though the facts can get blurry from time or obfuscation. Questions of purity are often emotional, value-laden judgements. At their best, I think purity questions are about historical accuracy. At their worst, purity questions tend to be about “protecting” an art from evolution, accommodation to cultural and legal changes, and questions about the purposes actually served by the art.
Thank you for explaining your take on that. It's interesting encountering these sorts of questions among modern 'jujutsu' systems; they don't seem to be much of an issue among koryu practitioners. Many of the koryu I am aware of incorporate modern variations or entire groups of techniques, and/or practice Judo in addition to their koryu practice.
Calling Japanese Ju-Jitsu a “hybrid” of these gets the order wrong, and thus is misleading about the scope of the arts. Perhaps my biology training is making me pedantic, but the parent isn’t a hybrid of the offspring. It would be more accurate (if, perhaps, provocative) to describe the descendants as (evolved, highly focused) subsets of the the parent art. If you’re simply trying to explain Japanese Ju-Jitsu to someone who is familiar with judo, aikido, and karate then it could be helpful to say that Ju-Jitsu includes the techniques found in each of these
Sure, but if you're going to be pedantic, one should begin accurately by noting that there is no such thing as 'Japanese Ju-Jitsu' - there are schools of jujutsu that still exist, but no single entity called 'Ju-jitsu'. Many koryu jujutsu schools do not in fact use that term to name their arts; instead using terms like kogusoku, kumiuchi, koshi no mawari, torite, yawara, taijutsu etc.
It would be more accurate to say that Kodokan Judo is a descendant of two koryu jujutsu schools; the Tenjin Shinyo ryu and Kito ryu, with further technical influence and development from a number of other systems. Judo does not have any real technical relationship with Aikido.
Aikido is a descendant virtually exclusively of Sokaku Takeda's modern art of Daito ryu Aikijujutsu. Of course Ueshiba Morihei and his family (and many others) have further developed Aikido as a collective these days.
Karate has no real relationship with Japanese jujutsu systems. As others have pointed out, Budoshin ryu is clearly an invention of Mssrs Kirby and Seki, based largely around Judo and Aikido. In fact it is exactly as described, a hybrid modern art. Which - again as others have mentioned - is fine provided one does not develop the mistaken idea that it is a 'parent art' of either of it's real parents..