Whoa, I never noticed those. How would he have no choice though? Like, forced to do it, or does he feel obliged?
I think it's more than a simple sense of obligation. Cole helps people because he wants to, but why does he want to? Because he embodies compassion. If Cole stopped helping people, he would no longer be compassion, no longer be himself. In a sense, he would have betrayed himself.
As for Solas, the elven people are obviously a sore point with him. One moment he denies any kinship with the modern elves and other times he regrets not doing more to help them. He admires Sera because she seems to have a purity of purpose that he lacks. Yet by the end of the game, one of the very last things he says is "But the people, they need me." Ultimately, Solas has to help them, however large the sacrifice required.
It's similar with the Inquisitor in some ways. You can deny the Andraste bits and act like a total jerk to everyone, but in the end you are the people's savior. There's no option to hop on the nearest ship to Antiva and spend your last days drinking brandy on the beach waiting for Coryfish to take over. The Inquisitor has a lot of power and influence over many things, but not over their ultimate purpose. (Short of not completing the game of course, but that's the player, not the character).