It's generally uneconomical to throttle output once the plant is built. because the fuel is so cheap. The real cost is building the plant and decommissioning it.

There is a lot of disinformation about nuclear power that has been so widely and consistently disseminated that it has basically diffused into the background.

no, you can do wild variations as long as it's not full shutdown

Some reactor types are better at going up and down in power fast than others. It is definitely a solved problem, though, and has been for decades.

I remember, decades ago, that anti-nuclear activists (some of them were even university professors who ought to know better) argued that it was impossible while France had already been doing exactly that for decades (at the time).