If I had to pick one essential profession it would be farmers. Even with peak efficiency it's work that must be done to sustain modern population sizes.

No, it's truly plumbing. We could not have any human gatherings close to what we have without plumbing.

People could grow their own food, eat straight nutrients if we had to.

Without plumbing you have no waste drain and no flowing water. I imagine you forgot about the irrigation aspect of farming.

Discovering plumbing, waste disposal and aqueducts[1] allow society to live in close quarters.

[1] https://en.m.wikipedia.org/wiki/Roman_aqueduct