Nuclear as used doesn't need utility-scale storage, but that's because it's almost always providing the base load with mixed generation. Modern European plants are sometimes load-following, but that's generally just two phases of high and low output. And even then, it's often more efficient to run at peak output 24/7, with cogeneration during low demand.
As far as I know, nuclear has never been used for peaking demand, but doing so would presumably require battery or pumped-water energy storage. (Since current nuclear is steam cooled, I don't believe molten salt storage is especially viable outside of theoretical high-temp plants?) Although pumped-storage hydro doesn't necessarily look much like damned-river hydro, so I don't think it's a great comparison there either.
That said, the gulf between where we are now and "nuclear as all of base load and some load-following output" is absolutely massive, and I'm not sure anyone actually intends 100% nuclear generation?
But this assumes 100% nuclear. Even a couple of fast reacting natural gas plants (or hydrogen or biodiesel or anything fast reacting) solves this.
Nuclear's on-demand ramping does actually seem to be between coal and gas, but my impression was that actually doing that absolutely destroyed cost-efficiency, to such a degree that it's usually better to just run at optimal efficiency and do whatever you can to recoup some money on the waster power.
Of course, we totally agree on the last point. There's room for major nuclear scaleup even in pure base load, and with a nuclear-heavy grid in mind it's not hard to work out a few peak sources.
But also nuclear has at least the flexibility to do it when needed [0]:
German Isar 2 nuclear power plant holds the record among currently running plants in adjustability. The reactor's nominal output is 1,400 megawatts, and it can be adjusted in a range of 400−1,400 megawatts, with an adjustment speed of 40 megawatts per minute. The need for this great an adjustment capacity has resulted from Germany's current energy policies.
[0]:https://www.lut.fi/web/en/news/-/asset_publisher/lGh4SAywhcP...
It was built to pair with nuclear power stations on the Connecticut river, specifically, to pump water at night during low demand, and then generate during high demand.
I suspect they're confused about what exactly requires the pumped water though.