Under what circumstances would a faction have the political wherewithall to attain its objectives through the Convention Method when it does not have the political wherewithall to attain its objectives through the Congressional Method?
We can all argue whether the 17th Amendment was a good idea or not. But at the turn of the century the Progessives made it a priority, and the people strongly agreed. The House agreed, but the Senate balked at changing the way it was elected. I've always seen this as a textbook example of what Mason was talking about.
The states began exercising their right under Article V to request a convention to address the issue of the popular election of senators. Had the Senate continued balking, a convention would have been called on the subject. But once the tally was one or two states away from a convention call, the Senate yielded to popular opinion and passed the amendment to the states of ratification. (I'm not going to argue the rightness or wrongness of that amendment here.)
Let me give you another more recent example. During the Reagan era, Congress was in no mood to entertain a balanced budget amendment, so the states began petitioning Congress for a convention to address the subject. The tally got within a state or two, but then the issue fizzled. Had it not fizzled, we would have witnessed the states taking control from Congress and addressing the issue directly via a convention.
Another example. I question whether an amendment extending the 14th Amendment to the unborn -- a human life amendment -- would be a good idea, but today you would have no chance of getting one through Congress. But the states through a convention? That might be more interesting.
Handling illegal immigration or the Kelo decision would also fall into this category. Do you see where I'm going with this?