Nuclear does neither flexible demand nor energy storage, those are in fact the things that nuclear does not solve! There are a few flexible nuclear plants in France but they push up costs. Some of the new modular nuclear rector designs are considering storage/flexibility, but cost there is also expected to be far higher than an AP1000.
Batteries are cheap, getting cheaper, and are the biggest disruption and innovation on the grid in more than half a century. You can use them to reduce transmission costs, to reduce the load on distribution substations and increase distribution usage capacity, you can use storage to make everything a lot cheaper by allowing smaller sizes for expensive T&D equipment that sees less than 30% average utilization.
Calling batteries "unrealistic" is not based in reality, it's just being stuck in decades past.
Nuclear can deliver flexible demand if required, it just involves either lowering the reactivity in the core, or if the drop in demand is sudden, bypassing steam from the turbine and running it directly to the condenser. But since their operating costs are so much lower than their construction costs operators run them at 100% capacity as long as they can.
The only situation where deliberately operating a nuclear plant at under 100% output is when nuclear makes up a very large chunk of a country's generation capacity. It's not that only French nuclear plants can reduce output it's that only the French have ever been in the situation where their nuclear capacity exceeds their minimum electricity demand.