On the specific topic of reprocessing though. Reprocessing achieves 2 things: 1. it can extract usable fissile fuel from spent fuel, and 2. it can reduce the amount of long lived radioactive waste, by a factor of 30. Point 1 can be further split in 1.a. usable fuel for the current generation reactors and 1.b. usable fuel for future, fast reactors (U-238).
1.a. Per wikipedia [1]
Reprocessing the plutonium into usable fuel increases the energy derived from the original uranium by some 12%, and if the uranium-235 is also recycled by re-enrichment, this becomes about 20%
In other words, all this reprocessing can reduce overall the total volume of uranium mined and spent fuel by 20%. That's not a game changer, and it certainly does not come for free.1.b. reprocessing in order to extract U-238 for fast reactors. That's a nice concept, but if we ever build fast reactors that can burn U-238 (fingers crossed), we already have a huge stockpile of depleted uranium. The US alone has more than half a million tons, and the rest of the world at least as much. That's enough to keep the lights on in the entire world for hundreds of years.
2. reprocessing in order to reduce the waste. That makes sense. But burying the waste is probably cheaper. We have already buried hundreds of thousands of tons of waste at WIPP. We know it works and it is safe. We know we will need to eventually bury some waste, even if its 30 times lower. If we, as a society, agree to open some deep geological repositories for nuclear waste, then it doesn't make all that much of a difference if we bury 10000 tons or a million tons.
[1] https://en.wikipedia.org/wiki/MOX_fuel
[2] https://en.wikipedia.org/wiki/Waste_Isolation_Pilot_Plant
There is, we know what it is, but no one wants to pay for it - it's called deep borehole storage. You basically put your fuel in cylinders, dig boreholes few miles deep, put all the waste at the bottom, done. It's not coming back ever, it doesn't pose any danger to anything or anyone on anything other than geological timescales(and if the geology starts pushing stuff buried 5 miles deep to the surface you have other bigger problems to worry about).
It's estimated that "just" 800 boreholes would be enough to store all nuclear waste ever produced.
But yeah, cost is one thing, but the other is that this material then becomes truly and irreversibly irretrievable - which actually might not be desirable since we know that even spent nuclear fuel can be reprocessed to make more fuel or weapons, which is not something that countries like US would want to dismiss as a possibility.
And also - storage of spent nuclear fuel is actually a lot safer and less scary than most people think. In Netherlands they literally made a museum out of their spent fuel storage facility, you can walk in between the casks that hold material radioactive enough to kill you in minutes, it's that safe.