... If you are 20 and broke when your captain tells you that he is going to sail into a hurricane on a rotten boat and you can quit now if you don't like it- you say, "Man. I need this job. I don't know how else I'll get home. And if I do, they'll know I bailed out and nobody will give me another TS crew gig."
When you are forty, you say, "Captain, I want you to reconsider. I'm happy to crew for you but we need to find a better course or else I simply cannot go. I love this ship, but I will not die for it. It's a great boat and all, but it's a boat. And it's insured."
I guess you are missing a point. Maybe if he was a sailor he could just say: "the hell with it, I am not going to risk my live, I quit."
But he is second in Command, tried to change the Captain judgement about sailing, he knows that he is going to sail with or without him, he knows that the crew revers the Captain and that it will go with him no matter what he does. He knows that he is one of the few with experience and that the chances to make it without him would be a lot smaller and he risks his life to give a better chance to the crew.
As Nolaton had pointed out, if the pumps were working 100%, if they had no problems at all, if they were lucky, as they were before, they may even have gotten away with it... one more time.... and his presence could be determinant to save the crew.
The time to quit would be after contributing to put the ship to safety (it turned out impossible) or immediately after the accident. I don't know if he was pressured not to do it, I don't know how much involved he was with that romantic connection with the Bounty's "family" and crew. It is hard to judge others when the involvement is not a rational one but an emotional one: They all considered Bounty to be their home and they were in love with the Ship.