The point of sending them off to a carefully selected globular cluster would already be to give them a better life.
I'm not saying someone won't convince everyone on board that a silver future for their Nth-great-grandchildren worth derailing the golden future of their 10Nth-great-grandchildren (especially if it looks like the generation ship won't last 10N more generations), but from my POV sitting on a couch that isn't on a generation ship (well, not something traditionally defined as a generation ship), it doesn't feel like something you'd get everyone all fired up over.