One is that nuclear plants are, among other stuff, massive heat engines. Because all the steel, tubes and whatever expands when it is heated up, switching it on and off stresses the material. This can be improved on by design but such design has extra costs and has its limits.

Yeah, and this is something that has been improved on for modern reactor designs precisely so that they can operate in load-following mode. There’s essentially no impact on operational lifespan (typically 60 years for modern reactors), because the impact has already been factored into the operational lifespan.

The second is that when you turn down your plant to half the output, you spend essentially the same money to get half the result. Which means you have just doubled the cost per kilowatt hour. And this with the background that nuclear is not any more cost-competitive to begin with.

This is mostly an opportunity cost thing. The actual running costs, e.g. the fuel, make up a negligible part of the €/MWh of nuclear. Most of the cost comes from the construction of the plant, which should be publicly subsidized the same as other clean energy is. Lack of subsidies and other public support is one of the main reasons nuclear is relatively expensive, though it is still the cheapest ecological method for meeting base load that we have.

In the result, a fleet of wind power plants plus battery or hydro storage is cheaper than such a nuclear plant.

The thing about battery storage is that it doesn’t exist yet and may never exist in an economical way. Hydro power and storage, on the other hand, is absolutely devastating for ecosystems, clean though it may be in terms of carbon emissions. It would be preferable if hydro dams did not exist. Now of course you could build a hydro storage system in a completely artificial pair of reservoirs, but that will be incredibly expensive compared to natural reservoirs (read: flooded valleys) so I am skeptical that it would be feasible at scale.