Launching waste into space with a rocket is a silly idea. We don't really care if the waste gets damaged as long as it doesn't break up and make a mess. Why not build a giant railgun to launch it?
I'd consider them equally as silly, considering we're actually capable of launching rockets into space yet the closest we've gotten with a space gun is a measly 180km apogee @ 3.6km/s[0]. The original question was why we don't send our nuclear waste into space. We can't send our nuclear waste into space using a rail gun because we can't currently send anything into space with a rail gun.
Cause someone murdered the leading scientist in the field. Also there's concerns that it'd be too easy to turn it essentially into a very long range, highly effective weapon that could lob nuclear or conventional projectiles at countries on the other side of the planet with little to no warning.
We already have highly effective weapons that can lob nuclear or conventional projectiles at countries on the other side of the planet with little to no warning: ballistic missile subs.
It's cheaper and safer to store it in deep crystalline bedrock. I guess if we geologists can't convince we the people of that maybe the railgun idea would work, especially if you could hit the sun where the fast fusion neutrons would consume the hell out of it.
Otherwise here's a good discussion on the viability of space guns: https://space.stackexchange.com/questions/2370/what-technolo....
[0] https://en.wikipedia.org/wiki/Project_HARP