that'll show em.
that'll show em.
(Setting aside the risky problem of getting this waste off earth and into space to begin with)
If you "tack" your solar sail, you'll spin.
It's similar to a car going down the freeway at high speed that needs to make an immediate 90 degree right turn. It can't do that without slowing down first, if it did, it would slide sideways off the sideroad. It's harder in space because the car is going 30 km/s and there's no friction that slows you down.
Yes, gravity is holding us back from yeeting of into space but to get closer to the sun you need to slow down. A lot. It takes way less rocket fuel to speed up enough to leave the solar system than to get to the sun.
https://upload.wikimedia.org/wikipedia/commons/9/93/Solar_sy...
Further, if you do decide to launch it, it takes significantly more energy to shoot it out of the solar system than towards the sun. [1]
[1] https://www.forbes.com/sites/startswithabang/2019/09/20/this...
The delta v to the Moon is something like 7 times less than to the Sun (3 vs 20ish).
The delta v need of the Parker Space Probe, which still "only" gets to 8.5 solar radii is so high it will use 7 Venus assists to get it close enough (it can't do big Jupiter assists because the solar panels it would need at Jupiter wouldn't be able to fit behind the sun shield at perihelion, and they didn't want to give it an RTG because they're saving plutonium for future missions).
You can eventually hit the Sun with enough Venus assists and Earth assists, for a total delta v of under 4 km/s, but it'll take a very, very long time, and your nuclear waste will be doing Earth flybys until finally it hits the Sun. Also it might be tricky to get the ball of sun-melted radioactive slag to do accurate assists after the sun melts it on the last few close encounters.