But how are they born? I'll have no fruit flies in the house, and then I'll forget a potato, it rots, and then I'll be infested with fruit flies. Were they already on the potato before it rotted, or did one just happen in, found some food, and figured it would stay and start a family here?

This is what I always wondered about maggots, like when a person in any kind of sealed room, anywhere, dies and starts to decompose. I've been lead to believe maggots inevitably appear and start the process but where did the larvae come from?! Food?

I think they’re actually on fruit or may be vegetables in your case from the start.

Yeah, that's my suspicion as well, as I rarely just see one lone random fruit fly in the flat.

They ride in on your grocery store produce. They also seem to hang out in/near soil in house plants.

How are those products shipped and stored prior to you purchasing them?

Yep, that's right. That's when the flies infect the produce.

Clean/peal the potato and immediately put it in a sealed container. No flies will appear. Only flies beget flies.

IIRC, I read in _The Violinist's Thumb_ that fruit flies are a more recent import along with tropical fruits (I think bananas?), so the spontaneous generation theory would predate their presence in the West.