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A small nudge near the outset would produce a massive ultimate change in’s trajectory, so maybe not too much energy is needed to redirect.

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Yeah, but why. It's not like you could redirect it enough that you'd land somewhere in your lifetime. It'd just be changing where the ship ended up in 10, or 100, or 1000 generations.

To give your descendants a better life?

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.

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