David, I'm not familiar with a single dojo going independant, but I was peripherally involved in setting up a new association a while back, which may be similar.
Essentially, what you described concerning non-affiliation etc. is correct. However, for insurance purposes, new associations/dojos normally do then join some sort of umberella organisation to ensure they are covered legally and have some sort of recourse if problems arise in the association and also for dan ranking. The organisation they join will normally be one of two type.
1. A generic martial arts council with fairly strong connections to some sort of government body.
2. An aikido asoociation which is best described as a cooperative "bank" of dojos.
In both cases, "style", syllabus, kyu gradings etc. are up to the individual dojos. It is only if the dojo wishes their students grades (and their own) to be recognised outside that dojo that stronger links are needed.
I was part-time in a dojo which seemed to be run as a separate entity as it switched associations three times in as many years. However, as gradings were not my interest, I found no problems with training there.
Hope it works out for you