You haven't answered the question though. How do you get the public on board?
It'd help to stop repeating mostly false stories (and leading questions) about Chernobyl, Three Mile Island, and Fukushima every time nuclear comes up.
But how to persuade the general public that science and statistics matter when even HN is skeptical? You're right, I don't know how to do that.
Science works because you are able to control most of the variables. It simplifies the world into something that can be modelled. Major decisions like energy policy are just more complicated than that. It is hundreds of overlapping considerations that have no good single answer. Yes science still matters, but there are deeper political questions at play. You just need to accept that if you want to build big complex things in the environment.
Of course these kind of issues can often be ignored with sufficient political support. Which is exactly what wind and solar have done in Europe.
Try using science to prove that a wind turbine isn't ugly. Or that it is worth killing some bird of prey to get clean energy. Try using science to explain why a coal miner should lose their livelihood. Or why you should stop eating so much red meat. It does not work.
At the heart of this debate is not scepticism of science, but scepticism of scientists and engineers. Scepticism that they really are as good as they say they are.
Fukushima required active cooling and could not cool itself using its own power output. And the backup generators were below sea level and the thing was on a fault line.
Neither caused major loss of life.
Three Mile Island was a scare, and did not cause significant damage or loss of life.
Any newly constructed reactor would not have the design flaws of the Soviet RBMK and would be designed to fail-safe.