I always thought the biggest reason was that if a rocket carrying nuclear waste exploded in our atmosphere, it would be catastrophic. It's a lot less risky to just bury it underground and pray it never leaks.
The author acknowledges the existence of that argument, but then insists that the "real objection" is difficulties in orbital trajectories. I'll believe that's the real objection if we start considering sending nuclear waste on a comparably simple orbital journey into outer space.
If you're going to bury it in the ocean, your best bet would be to bury it near a subduction zone. But then you have the earth itself possibly being the cause of the container cracks and leaks.
I ask because I genuinely don't know. I'd be OK with it if scientific consensus and consultation between the ocean and nuke people said we know enough about them to say there aren't issues. I don't want to just say "lets just dump them in a place I don't personally care about that's really far from me".