Nuclear has this problem too (in a lesser form though).
Most reactors mainly run at 100%.
Regulating them down takes quite some time and also wears down some materials inside.
It’s mostly a cost problem, idling a reactor for 12 hours produces 1/2 the power per day at 95% of the cost. Aka your cost per kwh basically doubles.
France largely got around this by exporting power, but that obviously doesn’t scale if everyone tries to export power on the weekends nobody wants to import it.
Finding something to use the power flexibly isn’t enough, they need to be willing to pay several times more than Solar for it or you aren’t fully offsetting costs.
Nuclear is fine with load following. Most plants run as baseload because it makes sense for them to do so, given the surrounding energy mixture and capital costs, etc, but it's not some sort of immutable law of the technology. Many existing designs have that capability even if it's not used much. They're not great for peak generation, but if you're price insensitive, or could lower the capital costs enough for it to make sense, they could do that too.
Regulating them down is done by more aggressively cooling the reactor. The thermal energy output is unchanged, but less of that thermal energy is transformed into electricity. This doesn't wear down any materials, but it does waste fuel. Grated, fuel is a tiny faction of nuclear's cost. Furthermore, the gap between peak and minimum electricity consumption isn't that large. It's usually 10-20%.
Which type of reactor are you talking about here? I’m not a nuclear expert, but the wonderful Chernobyl TV series doesn’t align with this. That reactor put out 3mw usually, peaked at 33kmw, and was idled down to 700kw using control rod insertion rather than more aggressive cooling.
Any reactor. This modulating is done with the cooling system, not the reactor itself. To clarify, the thermal output of the reactor remains unchanged, while aggressive cooling reduces the electrical output. It's deliberately reducing the efficiency of the heat engine to reduce electrical power output with the same thermal power output. That's why it's wasteful of fuel. Modulating the thermal output of the reactor is more efficient, but not all reactor designs can do this quickly.
The blast at Chernobyl was due to a positive void coefficient that meant over-heating led to a positive feedback loop (plus some other process failures that led to that initial overheating). Also those figures for the Chernobyl reactor were for thermal output, not electrical output. The figures are an order of magnitude off, the reactor would normally put out 3GW, not 3 MW and was idled down to 700MW for the test. 700 kW is, like, a beefy muscle car.