For any of those reasons, it may make sense for a generator to pay someone else to temporarily take their excess capacity, rather than shut the generator down.
I don't know how things work in Germany, but I'd be surprised if the negative prices there had much to do with the costs of starting/stopping conventional units. In most electricity markets, participants have to bid in their marginal cost. Even if you're an inflexible nuke, your marginal cost is positive. You usually need genuinely negative marginal cost units to drive the clearing price negative. That only really happens when subsidies are part of the picture.
Only variable costs will drive the short-term decision of whether to stay on or not. But the price of electricity in relation to fixed costs and profit will factor into the long-term decisions regarding building new plants and decommissioning ones that need refurbishment/upgrades. In the long run, the price of electricity has to be high enough to ensure sufficient capacity to meet demand.
(That's something of a simplification, as I said. In reality, the incentive for building new capacity comes from both the electricity market, and so-called capacity markets: https://business.directenergy.com/understanding-energy/manag...)
> What I don’t understand is why prices must go negative. A price of zero should be enough to turn off capacity, modulo switching costs (the cost to start or stop a turbine?)
It can take a coal plant ~4-8 hours to reach maximum output after even a warm restart.
The only type of cost that strictly determines a price floor is the variable cost. Recuperating fixed costs and profits are based on a revenue prediction which may or may not come true, but once the investment is made it rationally operates until prices descend past the variable cost barrier, regardless of whether long term ROI is positive or not.
What I don’t understand is why prices must go negative. A price of zero should be enough to turn off capacity, modulo switching costs (the cost to start or stop a turbine?)