That said it still seems better than many alternatives
We know what to do with it. Bury it, deep and somewhere remote. The US already has such a place: https://en.wikipedia.org/wiki/Yucca_Mountain_nuclear_waste_r...
Also, what about Not the United States ? It seems everybody is synchronizing policies, if you hear the rumblings out of the European Commission. Where are they gonna store the wast ?
https://en.wikipedia.org/wiki/Onkalo_spent_nuclear_fuel_repo...
The stuff stored in these facilities is not magic. It doesn't get up and run around. The sites are selected to be deep enough and to be resilient to leaks. I'm more concerned with our culpability for melting the world's glaciers and ice caps than the risk of someone digging up barrels miles deep a thousand years from now.
On your other info : I don't know about culpability. I'm passing the 40 year old mark, and I don't remember my generation, or the generations right before mine having enough influence on such matters until very recently.
These choices were made much earlier, by within your framing I'd say more culpable age brackets, and which are slowly let's say "disappearing".
What remains, in my opinion, for people currently in charge of affairs, and in the future, is a matter of responsibility to all future generations. We most certainly know what's what now, and the ethical and moral calculus is publicly in evidence, as a consequence of, amongst other such operations, the Greta Thundberg UN tour.
Again, not arguing against nuclear. If we do it, I'd say let's not create new problems out because we're too sure of our probability projections... As pointed elsewhere, we were due for an Ice Age. That might still happen down the line, whatever happens to surface human politics. Let's make sure somebody's there to check for such leaks down the line, and that it's easy to access.
Once we decide we won't make matters worse now, why stop at that ? Let's make sure we don't make them worse down the line out of some new error.
https://en.wikipedia.org/wiki/Lake_Karachay
https://www.vice.com/en/article/vbn9e9/the-soviet-union-dump...