As Delta V was saying in another thread, "And while I realize many of us here are deeply suspicious of political ideologies (myself included), I think it'd be nice if games started to actually occasionally espouse them."
@Fapmaster5000, the explanation you suggested in that thread fits the character and themes quite well: that Elizabeth, at the peak of her anger at the Founder's regime, and influenced by Booker's cynicism, looked for an alternate reality in which Vox Populi had successfully armed itself. Her anger unconsciously influenced her choice, and she selected one in which Vox Populi was successfully revolting, by especially violent means. That's consistent, for instance, with how Elizabeth begins expressing guilt, and argues that she and Booker bear significant responsibility for the death and destruction they are witnessing.
It may have made the story a bit better if this were alluded to: if for example Elizabeth speculated that the Vox could have prevailed with less violent means, and Booker argued that any revolution is violent, and that faced with things like the mechanical patriot, the Vox would have to have been brutal in response. That would have left us uncertain, while still reinforcing the sense of how dangerous this power was, and the temptation to use it. That's something that could have been accomplished with a few lines of dialogue — and I wouldn't be surprised if they were effectively in there already, and I just missed them.
But what really bothers me is, as Delta V suggested, a loss of an opportunity for discussing ideology, and in particular, for me at least, this one particularly rankles, because I see an all-too-familiar caricature dropped in, where I was hoping to see something that at least resembled the political movements of the time. Our discussions of ideology are often grossly impoverished by historical ignorance and by a tendency to lump together opposing views and to simplify them. This is the sort of thing that can be healed by good storytelling, and exacerbated by simplistic morality tales that reinforce preconceptions.
With what we see of Vox Populi, we're left with, at best, part of a story. Columbia seceded from the US, because the Founders favored aggressive imperialism, and intervened in the Boxer Rebellion, angering the US president. And Bioshock: Infinite takes place in 1912, not long before World War I. One of the most important things to know about the history of the American radical left is that the Socialist Party in the US, unlike most others, remained fairly unified and in opposition to the war on the principles of international solidarity and opposition to imperialism; it faced a wave of political repression, including the imprisonment of Eugene Debs, for agitation against the war. But do you get a hint about any of this from the game narrative? It's not like you can just assume that everyone playing the game has read Howard Zinn.