Once it's down there it might be safe, but the devil is in the details. How do we get it there with 100% reliability? If you think an oil spill is bad, consider the effect of a spent nuclear fuel spill.

Nuclear waste is mostly solid and very dense.

Water absorbs almost all harmful radiation within a few meters

The thought process behind just throwing barrels into an ocean trench as was done by the US was and still is sound but people are freaked out by the notion. The UK was encasing barrels in concrete before doing the same which is more than good enough by most measures.

Ultimately nuclear waste is easier to deal with than some other toxic industrial waste or chemicals.

How do you spill a ceramic?! It's a heavy solid, not a liquid!

But then where would we get non-irradiated steel from? We're already harvesting it from old, pre-nuclear-era sunken ships: https://en.wikipedia.org/wiki/Low-background_steel

Irradiated steel comes from irradiated air used in the steel making process, all of which came from bombs detonated in the atmosphere.

Nuclear waste sitting at the bottom of the ocean will have a hard time ever reaching the atmosphere.