Why we don't shoot Earth's garbage into the sun

Why not? There are currently three barriers to the idea:

The possibility of a launch failure. If your payload is radioactive or hazardous and you have an explosion on launch or during a fly-by with Earth, all of that waste will be uncontrollably distributed across Earth.

Energetically, it costs less to shoot your payload out of the Solar System (from a positive gravity assist with planets like Jupiter) than it does to shoot your payload into the Sun.

And finally, even if we chose to do it, the cost to send our garbage into the Sun is prohibitively expensive at present.