There have been numerous Jacobite rebellions, and The Royal Stuart Society still exists (although admittedly I’m ignorant as to their level of activity).
Is this a natural extension of the long-standing push for Scottish independence, or is there another factor that I’m missing?
Before you seek a comprehensive answer to this question, it is important to qualify what we're talking about when we discuss the Jacobite movement in context. To be clear, historical Jacobitism is long-dead and shares little with the beliefs of self-professed modern 'Jacobites'. This latter group tends to be largely made up of insular, republican, conservative monarchists who pine for the 'glory days' of Divine Right rule while performing flaccid rituals (toasts, speeches) imbued with sorrowful emotion and 'lost cause' conditioning. Some have appropriated the visual, cultural, and symbolic mechanisms of historical Jacobitism to reinforce their modern, often atavistic beliefs, but I have rarely encountered a legitimist who has any real grasp of what the Jacobite cause actually represented to the people who espoused it during its hundred years of activity, roughly 1689-1789.
I have previously summarized the primary tenets of historical Jacobitism here, so I won't fill this space with a repetition of it. But I would like to specifically address the issue of Scottish independence within the broader umbrella of the many Jacobite causes in the long eighteenth century. To put it plainly: Jacobitism did not equate to nationalism, which makes its appropriation today by independence-minded folks somewhat odd. While it is true that for some, support for the Stuarts signaled an alignment with rethinking – not necessarily revoking – the parliamentary Union of 1707, the regnal union of crowns was always going to be jointly ruled by James VIII/III if his son was successful in overthrowing the established Hanoverian government. When faced with widespread resistance from loyalist Britons (that's English and Scots) in and after 1745, Stuart doctrine justifiably promised pretty much anything that might attract a broad base of followers to get warm bodies on to the field – including the dissolution of the parliamentary union for those who thought it might be a more promising option. The resistance to the Jacobite threat, however, was also made up of large numbers of Scottish patriots who thoroughly felt the union was beneficial for Scotland, even through its growing pains and admittedly uneven advantages. They also felt that Jacobites were by their very beliefs committing treason against their king and country. So we see that nationalism cuts both ways in this context.
To sum it up, taken from my linked post above: "... independence-minded nationalist sentiment was a relatively small motivating factor in the bottom-up constituency of Jacobite supporters during the last rising. The Jacobite-Nationalist alignment has been subsequently force-fit due to the hot potato of the modern independence issue, and historical Jacobitism in the context of the eighteenth century has been distinctly misshapen by those in privileged positions who, in my opinion, should know better. This is certainly not a commentary on the viability of the nationalist agenda or the legitimacy of an independent modern Scotland, but a frustrated complaint that this vital issue should not be conflated with Jacobitism in a widespread historical tableau. They weren't and aren't the same things, however present a small degree of overlap."
Finally, to your question about what makes the Jacobite cause so enduring: as mentioned above, there is a distinct romanticism connected with the historical Jacobite movement, largely attributable to the 'fateful doom' or 'lost cause' character of that history seen through the antiseptic lens of the modern. It's true that horrible things were done to innocent people in the aftermath of Culloden, and those measures arguably strangled the last gasp of the feudalistic Highland clan system, perhaps already centuries out of its time. It's also true that terrible and particularly emotional things occur in civil wars, of which the Jacobite challenges were certainly a part. Brother against brother, pomp and tartanry, Bonnie princes and all that. The symbology is indelible and was designed to be that way specifically to attract followers to the cause.
But why does it endure? It might be uncomfortable to ask the same question about the Confederacy in the American Civil War, but the answers, if not the causes, share a quality and the same ultimate fate: progressiveness won out versus restrictive, elitist atavism. There is a fine line between rebellion and revolution, that line being a simple matter of failure or success. Perspective and perception is essential, then, when we discuss terms like independence, loyalist, rebel, and patriot. But the grass remains ever greener on the What If? side...
Some relevant titles for further research:
I hope this has been of some help.
With best wishes,
Dr Darren S. Layne
Creator and Curator, The Jacobite Database of 1745
(recently of the Institute for Scottish Historical Research, University of St Andrews)