Everything you said makes perfect sense - unless you spend two minutes learning the most basic facts about nuclear waste.
There are basically two kinds of radioactive waste.
There is a small amount of highly radioactive waste. Highly radioactive means it emits it's radiation quickly. That's bad because it emits a lot right now, and good because because by emitting it quickly, the radiation is gone pretty quick. That stuff you want to store in very thick steel containers for a hundred years or so. Since there isn't much of it, that's no problem.
Then, there's the stuff that emits its radiation slowly, so it lasts a long time. On the other hand, because it is emitting slowly, you'd need to have it in your house for a few hundred years before it would make you sick. As a demo, I was going to eat a spoonful of it, which would probably give me a belly ache similar to eating an entire pizza. So no problem with that part either.
The only problem with nuclear waste is that some people don't know the difference. It's purely a political problem, there's no engineering problem. Safe storage is easy, technically speaking. Getting people to understand that after 50 years of misinformation by the anti-business lobby is the hard part. You'll notice that the exact same environmental organizations and leaders who convinced you that nuclear was bad are now trying to undo their earlier misinformation campaign, nowtthat it's obvious that nuclear is the only workable alternative to petroleum for 90% of our needs. Most environmentalists have realized that they can't continue to pander to their traditional allies, the purely anti-business constituency. To save the planet, they have to leave those old allies behind and tell the truth - for most of our needs, it's either nuclear or petroleum, and nuclear is by far the better choice.