Yeah I'd like to visit these gyres and see the mess but I think it would lead me to join an ultra-violent arm of Greenpeace after seeing something that horrific.

How exactly do you think countries can deal with waste from nuclear reactors? It can't be destroyed, only buried or moved. Maybe eject it into space?

The problem with the way countries are dealing with it now is that it will create problems for future inhabitants of the planet. Given our propensity to dig and see what's buried in the ground below us, 10000 years from now when evolution has humanity's progeny back on top of the food chain, they will dig and find one of these caches of nuclear waste. Unless they can read English or one of the other languages used on the warning signs or they're intelligent enough to have Geiger counters on hand, they'll unearth lethal radioactive material, possibly unknowingly spreading it and killing lots of people.