4 ideas
A mass rebellion concludes the predicted helium flash was a hoax used to justify generations of suffering, overthrows the Coalition Government, and leaves the few thousand loyal engineers and scientists on the open ice to freeze to death. Soon after, the sun undergoes the helium flash exactly as predicted and destroys the inner solar system, proving the executed experts right after it could no longer help them.
A danger that has not yet happened cannot be seen, but the cost of preventing it is felt every day. The longer the sacrifice lasts without visible proof, the more the lack of disaster looks like evidence the danger was never real.
The trip to Proxima Centauri takes about 2,500 years, so no one who starts it will see it end, and no one who arrives will remember why it began. Each generation's role shrinks to keeping the machinery running and passing the purpose on. Meaning has to come from the relay itself rather than from reaching the goal, which is exactly why belief in the purpose can break.
Under a species-level survival emergency, marriage, romance and even grief for parents become casual and temporary: couples part easily, and deaths are absorbed without ceremony. Read this way, the depth of personal bonds depends on a stable future that can hold them. When the collective future becomes the only thing that matters, private feeling is reclassified as a luxury.