Admittedly I have never used them as written in a BTB campaign because my group uses a different method of determining opponents than suggested in BTB or MiM. So my answer may not help, but when I tried adapting BTB objectives to our Pyrates of the Karribean games, the campaign took about as long as it did without objectives as it was hard for players to fulfill the objective in the first place. Then when they had, it was hard to win the final scenario against the other players who had stayed in the campaign.
Short answer 20+ games per player.
Out of five campaigns we used objectives in 3 ended with just declaring the player with the most campaign points the winner.