On closer inspection, everything requires a name. I'm guessing this prevents computation over spurious objects?

Fyi: the concept of using hard coded rules to define areas will quite likely disappear later this year, meaning that you will be able to treat every closed object as an area.

Would it be possible to add power=plant as a interim fix until the code is refactored?

I’d say it’s somewhat unlikely to see this in production in the near future. Probably you need to look into setting up your own overpass instance, or use some alternative approaches.