Jujutsu is Aikido, Right?

I was corresponding with someone in the Jan de Jong tradition who made the very thought-provoking comment: 'Jujutsu is aikido, right?' 

I don't know whether their comment was serious or not, however, it does raise an interesting issue. One that should be a part of any theory grading for jujutsu and/or aikido, if those grading systems have theory gradings (as Jan de Jong's gradings do).

This issue is brought into stark relief with Yoseikan ryu and Gyokushin ryu. 

Mochizuki developed/created Yoseikan ryu. He originally trained in Gyokushin ryu jujutsu, which is the subject matter of the only book he published. The methods included in that book formed the basis for what became Yoseikan ryu aikido. But there are some who teach those same methods under the name of Yoseikan ryu jujutsu.

Some have reverted back to the Gyokushin name teaching either Gyokushin ryu jujutsu or Gyokushin ryu aikido.

When does Gyokushin ryu jujutsu become aikido, and when does Yoseikan aikido become jujutsu?

Mochizuki's teachings had a huge impact on Jan de Jong's jujutsu grading system. Much of that influence formed the basis for his aikido grading system, however, a lot of the jujutsu grading system that is not Mochizuki influenced did not make its way into his aikido grading system. This is a good example of - is aikido simply a subset of jujutsu (albeit some jujutsu systems; for instance, there can be no Brazilian aikido related to BJJ, although I do not put it passed someone to attempt that)?

This contemplation raises so many issues. Can a particular qualified jujutsuka develop an aikido system but the opposite is not true. After all, that is the theory of the development of aikido.

Are they, as the JDJ tradition member suggests, one and the same and the change of name is simply a 'rebranding' in order to 'differentiate the product' (marketing 101)?