Did Gaddafi have majority support during his rule, did that change to minority support ? If so, at which point did that happen ? And have we got reliable sources for any of this ?
I refer to support from within Libyan population.
When you live under a repressive regime you may appear to be supportive, but privately you realize you don’t really have a choice. Once the tables started to turn, and the rebel alliance was able to make steady gains against Qaddafi’s military, more and more people said what they really thought about him publicly. So I would argue that he never really had ‘true’ majority support, it only looked that way because people lived in fear of being imprisoned or worse.
There’s an implicit assumption in the OP that someone could engender majority support in a society like Libya’s, either pre- or post-Gaddafi.
More realistically in such a broken society is a situation where no faction can command more than a plurality, and even then only in one region within the country.
The challenge is make the society homogenous enough, and successful enough, that the possibility at least exists for a majority to form and to have a stake in the status quo.
The US today is seeing its own majoritarianism under siege both from factionalism and from regionalism.