Farmer villagers don't really harvest quickly. If you're doing this for the carrots there are micro farms that are a lot quicker (but require bonemeal, and user input). The villagers usually throw food (carrots, wheat, bread) at other villagers in order to breed. I'm not certain if they have a item cap or not, but they dont just see grow crops and harvest them. If you REALLY want this to be a carrot farm run by villagers, there is a diamond shaped farm tutorial (technically a villager breeder) where you can put a hopper under the breeding villager, and it will collect a bunch of extra crops that the farming villager tosses. Honestly though, the most efficient/resource friendly version is just plant rows and rows and collect them yourself.