Why didn't the citizens of Polaris, Regulus and Sirius just vote the government out?
The citizenry did, broadly, support the return to Sol. That was part of the importance of the project - to mollify the populace and offer a carrot to work for. But the dangers the GTVA faced weren't as simple as failure of confidence in a single government or set of leaders. It was a failure of confidence in the system as a whole - the kind of unrest that isn't expressed through democratic channels. (Again, the kind of unrest exploited to create the NTF).
Moreover, look at the GTVA's lineage. The GTA was basically NATO - a military organization with very little power in the civilian sphere. The GTVA stands closer to a true interstellar government, but of all its branches, the General Assembly is the only one that can really be said to respond to public opinion. There are powerful civilian positions which would remain more or less untouched by elections and the like. The Security Council is, by design, stable.
Remember, this is a government that spans worlds. There are powerful positions in every democratic state on Earth which are intentionally isolated from public opinion. The GTVA has similar positions, both civilian and military - GTVI, for instance, is a powerfully influential organization.
The GTVA has existed for less than fifty years. Prior to its formation, there were multiple, independent Terran states, and those identities haven't been submerged. The GTVA bought the loyalty of the Lost Generation with the promise of security and expanded human power in the face of the Shivan menace. It couldn't sustain that promise, and that threatens its very raison d'etre in the public eye.
The issue is not that 'our leaders have failed us'. The issue is that the relatively young GTVA failed catastrophically in the first test of the very promises that created it. Or, at least, that's how people see it - a perception shaped by economic problems as much as by military realities.
The GTVA is a democratic state. So were the early United States. But the US faced enormous challenges across the first hundred years of its existence, and at no point in the first forty years of its life (which the GTVA still occupies) was anyone confident it would endure rather than splinter. It took the Civil War to really settle many of the lingering questions about the American political system.
I think that's a good analogy for why the GTVA's problems can't be solved by an election cycle.