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.

You're proposing to enslave a thousand generations to ship them off to dead worlds rich in certain resources. I have no idea how you imagine their life to be once they arrive, but I can promise you that you're not giving the first 2-5k generations a better life, you're forcing them into a horrible future. It's deeply concerning that you don't seem to understand this.