The disparity between peak energy load and minimal energy load is only ~25%. Excess energy is an easy problem to solve. Nuclear power plants' thermal output is largely fixed, but their electrical output can be modulated by more aggressive cooling. Basically, deliberately produce more waste heat. If this nuclear plant is on the coast it can use this waste heat for desalination - the waste heat gives you freshwater as a bonus.
But the same solution also makes solar interesting again when it's used in combination, because you can store heat from nuclear during the times when solar is generating and then use it during the times when it isn't.
If you remove fossil fuels, you have to replace them with something. You're building new power plants. New nuclear power plants could straightforwardly be built with thermal storage.
Nuclear plants definitely could use thermal storage, but the main advantage is of nuclear over renewables is that they have consistent energy production and thus don't need storage in the first place. It doesn't suffer from the duck curve like solar, or weather-dependent intermittency like wind. Nuclear doesn't need storage to become viable, as France has demonstrated for decades.