Could they overproduce and keep unsold stock for next winter, and if unsold stock gets too high, stop producing more until it reduces?

They mostly do keep unsold stock, only a fraction of it gets destroyed. See the EEA's full analysis from 2024 (https://www.eea.europa.eu/en/analysis/publications/the-destr...).

They could, but it’s a tradeoff. Inventory costs money and if you cut production, that means laying off workers and possibly selling productive assets, at which point it becomes more expensive to scale production back up.

Every business decision is a tradeoff. Smart government interventions in the economy add weight to that tradeoff to reflect externalities not otherwise accounted for; this is how cap-and-trade on SO2 emissions works. Hamfisted government interventions set hard and fast rules that ignore tradeoffs and lead to unintended consequences.