To my knowledge, apart from very new battery-based systems, the most common energy storage used for grid scale applications is pumped hydro, and even that is pretty rare... Either you need geographic features that make it viable, which is relatively rare in proximity to all the geographic features you would need for a nuclear plant, or you need to build such structures which is insanely expensive.

The main issue with grid scale anything, is that until very recently, most energy companies have been living on insanely long timelines, far longer than most industries. Infrastructure, when built, almost always has multiple decades of lifespan if not longer. Most energy storage tech that's old enough to be considered for the time that many of the nuclear were built, did not have multiple decade lifespans and would need full or at the least the majority of their working material replaced within a decade at best. For an industry where a new facility will last 50+ years, that's not a good investment. The only long term solution that would last is pumped hydro. This is changing and new grid-scale storage tech is reaching a high level of development, aka, almost ready for large scale production.

Simply put, if you think about the technology that was available when these facilities would have been built, around 50-80 years ago for many, energy storage wasn't something that people really thought about, you either had live delivery of energy (from generator to device in micro-seconds) or primary cell batteries, like alkaline. Without much in-between, and most of what was there wasn't grid-scale, not even close.

Sure, there are newer reactors than that, but a remarkable number of nuclear energy facilities are many decades old, most of the viable grid scale energy storage tech has been developed in the past decade or so.

To be clear, nuclear plants usually have conventional generators, often diesel, but that energy isn't for export (for sale to customers), it's used to restart the pumps and power the facility for a cold start of the nuclear generation systems. And that's about it. 50 years ago, you didn't have viable energy storage for the grid, everything was generated as it was used, when the load increases, fire up more generation capacity. So base load was handled by plants that needed to run 24/7 like nuclear, since it's difficult or impossible to turn them off, and they would ramp up when demand increased, and any gap would be filled by plants that can go from off to making energy in minutes, like coal and natural gas.

To summarize, with the exception of pumped hydro, nothing is capable of handling that much power for the grid. We're not talking a few minutes of energy storage, this is more like an hour+ as reactors heat up and more turbines come up to speed. The only energy generation that can meet that demand that quickly is coal/gas-based plants and pumped hydro, with pumped hydro being so difficult to build, coal and gas are used. Of the gas-powered plants, natural gas is the most economically viable.

This is changing, but the infrastructure in use is usually significantly older than the technology you're mentioning.