I have four identical fields with villagers farming one type of crop each. To get their food, I have a second villager placed so they throw food at him, which then is collected into a water stream. Every collection area uses the same design and each one of the eight villagers had an empty inventory when assigned to his job.

While the three couples harvesting carrots, potatoes and wheat work just fine, the one harvesting beet does not. When I do harvest by hand, the farmer picks up the seeds and the beet and replants. But he doesn't share his crop with the second villager and he doesn't harvest himself, probably because his inventory is full already.