I consider organizations useful for two reasons: 1. Support
2. Credibility. Everyone needs to be a student and have a teacher in order to continuously advance.
Can't argue with the first point. Organizations do offer support. In the eyes of a lot of people, they also offer credibility as well, but maybe that's because they take advantage of a flaw in human nature. If a lot people say that someone is good, then people naturally assume that this person must actually be good. So, why not get a bunch of your friends together, form an organization, and all call yourselves "soke" or 10-dan? That's exactly what many people have done, but most people here would not recognize those groups as credible or legitimate, although the average person might.
People here do recognize the major aikido groups as credible, but if you go to certain other Internet forums with other standards of credibility, such as martial effectiveness against an actively resisting opponent, you'll find that these same organizations are not held in such high regard. Even some people on this forum feel that way. So in terms of credibility, it all depends on who you want to impress. I think those who judge credibility by membership in an organization are abdicating their responsibility to think for themselves and reach their own judgment about an individual on his or her own merits. While I would say that there are some organizations in which membership would immediately call into questions one's credibility, I don't know of any organizations in which membership would make someone instantly credible to me.
Finally, being a student and having a teacher does not necessarily have anything to do with belonging to an organization. One can belong to an organization for political reasons without putting much faith it in its leaders as teachers. I know a couple of people who operate dojos and that by their own admission (privately, of course) fall into this category. One can also be a student of a teacher without being a part of that person's organization, assuming that person even has one. The student-teacher relationship is something that occurs directly, in person, between two people. It cannot be reduced to something written down on a piece of paper.