The other replies handle the large issue now, but let me go point by point.
> Why is nuclear the silver bullet that will save us?
Not supposed to be that, as others said.
> The waste will be around for longer than we are. We have no solution other than 'lock it up'.
If we sacrifice a Cubic mile to save the planet that's a god trade. Paul is a lot bigger than Peter here.
> Accidents do — and most like will — happen. How will increasing the volume of reactors not increase this risk?
The Tsunami was far worse than the meltdown; I am afraid people's fear-memory often conflates the two. I'm willing to admit the official stats with Chernobyl might be a coverup, but still, tons of poor people around the globe get sick or die from perhaps-illegal fossil fuel pollution. It's not a worse coverup than environment justice issues in the US.
The annoyance here among pro-nuclear people is that other thing's problems get to slip underneath the radar, but nuclear's don't.
> storage
Consider storage will require tons of lithium. The ecological damage from that mining will also be catastrophic, and the ecosystems might take as long to recover.
I'm not saying we shouldn't do lithium-ion batteries. We have little choice we're going to need more. Same thing with nuclear. Same thing with not decommissioning dams as soon as we like.
The fact is with all routes, avoiding ecological collapse will cause local environmental issues. What irks people is the singling out of nuclear, as if other things didn't have issues too, and other essential systems (like top soil) repair as slowly as some radioactive isotopes degrade.