It's a side effect of the economics of nuclear power.
The cost of the nuclear fuel is more-or-less a rounding error. On the other hand, personnel has to be paid no matter at what load the reactor is operating, and paying back the construction loan is also a fixed monthly fee.
This means running a nuclear reactor at 10% costs about the same as running it at 100%, so running it at anything but full capacity means you are losing out on potential income. Running it at 10% capacity means having to sell that electricity at 10x the usual rate!
Combine this with the incredibly high risks involved in such a massive long-term investment, and most reactors operate on a "strike price" principle: they negotiate a fixed electricity rate with the government, and the government has to pay (or gets to keep) the difference between the market rate and the strike price. More importantly, this applies to all the electricity the reactor could sell: if the reactor is able to generate power but the grid operator for one reason or another wants to source it somewhere else, the reactor operator still gets paid.
All this means nuclear reactors can only really be used to supply base load. In an ideal world we'd want to use them as peaker plants to make up for deficiencies left by solar and wind, but in practice we'll be turning off wind farms to let nuclear reactors run. Nuclear power might've made sense a couple of decades ago, but there just isn't a place for them left in a modern renewable-heavy grid.
As long as you stay under 5%/minute or so, yes, you can with the latest generation of nuclear plants. Above that there are other alternatives that are more suitable (pumped storage, natural gas fired turbines).
And indeed, what you describe is exactly how it goes, nuclear power pretty much dictates the price of energy because it is the most expensive form. This is the big controversy of market based energy pricing to me. It should be the cheapest energy source that determines the price, not one of the most expensive ones.
I've recently looked into this in some detail and the situation is actually a lot more complex than you might think looking from the outside in. This is an interesting article on how energy markets arrive at their price (it is incomplete, and a bit dated, but still useful):
The cost of the nuclear fuel is more-or-less a rounding error. On the other hand, personnel has to be paid no matter at what load the reactor is operating, and paying back the construction loan is also a fixed monthly fee.
This means running a nuclear reactor at 10% costs about the same as running it at 100%, so running it at anything but full capacity means you are losing out on potential income. Running it at 10% capacity means having to sell that electricity at 10x the usual rate!
Combine this with the incredibly high risks involved in such a massive long-term investment, and most reactors operate on a "strike price" principle: they negotiate a fixed electricity rate with the government, and the government has to pay (or gets to keep) the difference between the market rate and the strike price. More importantly, this applies to all the electricity the reactor could sell: if the reactor is able to generate power but the grid operator for one reason or another wants to source it somewhere else, the reactor operator still gets paid.
All this means nuclear reactors can only really be used to supply base load. In an ideal world we'd want to use them as peaker plants to make up for deficiencies left by solar and wind, but in practice we'll be turning off wind farms to let nuclear reactors run. Nuclear power might've made sense a couple of decades ago, but there just isn't a place for them left in a modern renewable-heavy grid.