So I did a bit of research on my train ride back on why we don't just dilute and spread nuclear waste. Apparently, the question if we should dilute toxic waste to get rid of it, has been discussed for ages and nowadays, every paper or article, I found on this just states that "dilution is not the solution to pollution" and then there is no further elaboration on that as if this statement is so obvious that it is not required to explain it. In one article, they even wrote, that diluting nuclear waste is like a reverse Midas, where everything he touches becomes not gold but toxic waste. I don't think, this is how dilution works. There might be binary thinking going on here, where the authors intuitively think that waste is either toxic or not toxic and don't consider if something is more or less toxic. 

It was quite hard to find anything where they didn't just state that "dilution is not the solution to pollution" and considered that sufficient explanation. Eventually, I found someone on Reddit who claimed to be an expert. I am aware that this is not the best source, but it was the best I could find before my train arrived. :)
He writes that diluting and spreading would work for uranium and plutonium left in spent fuel and also for fast decaying fission products, but this is not a good idea for other slowly decaying fission products like some cesium, and iodine isotopes, as those tend to concentrate in bodies of animals after being consumed and then gets passed up the food chain and gets concentrated further (similar to heavy metals, I guess). 

So what we would want to do is to separate these particular fission products from the rest and only store those in a concentrated form. And it turns out, this is actually exactly, what is being done in reprocessing plants. So I need to take back, what I said about reprocessing plants. They don't just concentrate nuclear waste, but separate the different kinds. Btw, the plutonium and uranium that is being recovered is then not diluted and spread but reused, which is actually better, I suppose. 

I am not quite sure what to make with this. I guess, there is in fact a reason why we don't dilute and spread radioactive cesium and iodine isotopes, but burying them all in one place and hoping that our descendants won't at some point drill into them also doesn't seem a good strategy. Probably the best thing would be to store them in a way that they remain accessible and in a way that we don't forget about them, until we found a way of breaking the atoms down further into isotopes that are not radioactive or have a short half life. Which is pretty much what we will probably be doing for the next 70 years in Germany anyway.