Additionally, even with no nuclear reactors whatsoever, we'll still have to deal with nuclear waste as it is used in various other areas like medicine. The Goiânia accident incident in Brazil killed many people _immediately_ (unlike Fukushima) and originated from a defunct hospital, not a nuclear power plant. We need to find ways to deal with this stuff regardless if you use nuclear power or not.
The experience of that has been a bit mixed in the real world at Windscale/Sellafield "Europe's largest nuclear site" in the UK.
The incidents section on Wikipedia goes on for pages https://en.wikipedia.org/wiki/Sellafield#Incidents
and "The site is due to be fully decommissioned by 2120 at a cost of £121bn."
which doesn't sound ideal for future reprocessing.
These repositories are meant specifically for the separated waste from the recycling/reusing process. So the process you're suggesting is the one that creates this "artificial problem".
Everything nasty from all of them put together is something like one large modern soccer stadium.
Here's a good paper on the costs https://www.mdpi.com/2076-3417/9/12/2437/pdf
It has being done for 20+ years now.