You are correct that they are not.
But this is due to policy, not physics. It's obvious to shut-down the fossil fuel plants first, but after then, there's no physical reason why they can't keep shutting power plants down.
Anyway, agreed, adding storage is a much better solution than focusing on the management of renewables.
Since we're being pedantic, as I understand it, the grid operators don't usually tell plants what to do (outside of system stress response, curtailment, etc), the grid operator shares the forecast, and when the price forecast is low, fossil fuel plants are likely tell the grid operator they'd rather shutdown than produce power at low/negative prices. For solar plants, there's no fuel cost, and there might be subsidies, so producing at a negative market price might still be positive for the generator and there's no reason to turn it off. For nuclear, fueling schedules don't really change based on use, so there's no reason to not provide optimum power outage other than for grid stability.
Grid scale storage should reduce price swings, since storage plants will tend to show up on the demand side when prices are low and the supply side when prices are high; although perhaps price swings will become bigger when prediction fails --- if storage fills up by noon you'll have a lot of excess supply until sunset; if storage empties by midnight, you may have a lot of excess demand until sunrise.
In Australia, this has been happening for some time - plants are literally having their output dialled back (or even being disconnected entirely) during peak times by the market operator.
[1] https://www.abc.net.au/news/2021-10-23/solar-farm-overload-h... [2] https://reneweconomy.com.au/aemo-slashes-output-of-five-big-...