No we shouldn't. Nuclear waste is not and will never be solved. Nuclear meltdowns are real. Also, nuclear is more expensive than wind and solar.
No need to worry though, on the current trajectory I doubt our civilization will have the opportunity to deal with it.
1. Any equipment or substance that is both nuclear and potentially dangerous has some sort of magical mutant super-power to stimulate superhumanly idiotic behavior in a non-trivial fraction of the people handling/operating/managing the equipment or substance. That behavior is almost always coupled with pathological denials - both from the mutant super-idiots and their supposedly non-mutant peers.
2. General population humans are quite aware of problem #1, hence extremely distrustful when any human experts or organizations try to reassure them as to the safety of anything nuclear.
3. That "reprocessing" ain't nothing resembling "put on a lead apron and pour it through a coffee filter". Instead, it's a quagmire of chemicals and processes that'd still be horrific if nothing nuclear was involved, plus weapon proliferation overtones, plus all the problems noted in 1. and 2. above.
1. Plant construction is very expensive, and not in small part because people who don't actually understand the technology take every opportunity to tie up construction in the courts and regulatory agencies.
2. General population of humans are quite aware of #1, so even though they know nuclear wins on a technology basis, they know it will be unduly expensive, preferring to knuckle under to the mob instead of doing the right thing.
3. People not familiar with the technology don't realize that while we don't generally do reprocessing in the west, it's primarily due to cost, and that Japan will start reprocessing 800 tonnes of fuel per year starting next year.
The technology is the safest way we have to generate power, and could be the lowest cost if the regulatory environment was adjusted properly. It's just a shame there are so many people who are irrationally opposed.
Vs. those with both the unquenchable zeal and bottomless resources to win a multi-decade crusade to prove that they're Ultimately Right (but no short-term need for electrical power) should embark on an epic quest...
I'm not saying there are no risks, but if we don't start making more nuclear power companies, Elon will and he will go from $1T to $8T. Who am I kidding, he's probably the only one that can turn them around in 2yrs with nextgen tech. Everyone else is going to keep faking it and build them in 20 years for 100x budget.
Not considering this tradeoff has led to much carbon waste.
the politics of the threat of proliferation kills it in the crib.
I am hugely skeptical that it will be solved economically given that it suffers from many of the problems currently pricing coal and nuclear out of the market.