What would happen, if you managed to make sure all this stuff got escape velocity, is you would wind-up with a bunch of junk (literally) in an orbit around the sun that would inherently pass through Earth's orbit - as long as it doesn't have solar escape velocity, once a thing is moving on pure momentum, it is in orbit and since the orbit fairly closely an ellipse so the object will return. As the article mentions, probably not when the earth return to the point. But if you keep throwing stuff "out there", the chances of a return are naturally going to increase.
its not a completely rubbish idea.
if you shoot the right way you could end up cancelling enough orbital velocity to intersect the sun before any return to earth.
Throw in reusable rockets and... it's still really dumb. We have reactors that can make use of "spent" fuel - they're just not legal.
Space elevator with a railgun at the top of it maybe makes sense.
Sure, but far less than de-orbiting into the sun. Use the rocket to get to the moon-based rail gun where there is no atmosphere.
Iff one can make a container that can survive the blast/lift-off -- one wouldn't need to worry about the launch vehicle exploding in-atmosphere: there'd be no rocket fuel. Added bonus: find a way to use existing war heads, and combine with nuclear disarmament...
A solar powered mass-driver could sling a steady stream of pellets from Earth orbit into Solar non-orbit using existing technology.
That being said, burying it deep near the the start of a plate subduction zone makes at least as much sense.