Contrast this to the matching problem, where the outcome is where ALL the candidates are assigned. You're allowed to have an opinion about where you are assigned. I'm allowed to have an opinion about where I am assigned. These opinions might conflict indirectly because of limited spots, but they don't conflict directly. This is very different from the election case, where if you and I have different opinions about who we want to win, those opinions are always contradictory.
Obviously once there are two candidates remaining, Gibbard's Theorem doesn't apply, so I'm guessing that any procedure which reduces the set to two outcomes must itself be subject to strategy, but it would be interesting if allowing people to cast a separate vote in a run-off election was enough to make the first round no longer require strategy.
Another way to see this: the entire process of choosing two finalists and then having a runoff to choose an ultimate winner counts as a collective decision-making process. That there are two separate votes doesn't actually matter. By Gibbard's theorem, then, if there isn't a dictator, then the entire process is strategic. Since the chance to use strategy doesn't occur in the simple-majority final runoff, by process of elimination, it must occur in choosing the candidates who qualify for the runoff.
Doesn't this mean it's actually not very applicable to voting systems?
Voters generally only care whether their preferred candidate comes first, not about the relative ordering of the other candidates.
I suppose a voter might care how close their preferred candidate came to winning (or their margin of victory), but that's still a slightly weaker requirement than having a total ordering of all candidates relative to each other.