The farmer will not plant carrot onto nearby exposed farmland (even if he had carrots in inventory). The village is auto-generated, feature 4 (one killed by zombie yesterday) villagers and more than 4 (6?) beds in a fairly stretched out formation.

The seed is 67. The village is the closest (0,?,0) one related to spawn.

The gathering point is roughly 7 block away from the farm.

The only work site is a stone cutter and (in picture) composter. Rest of villagers are unemployed.

Also, seems like the carrots traded in to him is not added to his inventory.

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Edit:

The villager will only plant carrots onto nearby farmland only when picked up carrots I dropped to him. Even though he gathers more than one carrot from a fully grown crop he still planted only once.