Should work if the brown robed villager is an apronless farmer type, farmer/fisher/shepherd/fletcher.
If you don't remove any villagers it should stop at 12 villagers.
Starting from just 2 could take a while, you could try starting with them in a smaller room and tossing them food and/or trading to make sure that they are close to each other when they become willing. Though I don't really have much experience, for all I know that may be small enough.
Personally I'd plant potatoes so the farmers inventory doesn't fill up with seeds (and I prefer potatoes to carrots.)
(If that does happen you can just remove all the wheat (?) and follow the farmer around removing everything he plants until he runs out of seeds and then replant.)
The design is viable but needs more light. (It also needs to be far enough from other villages, but I expect you've accounted for that…. )
The only light sources I can see are some torches on the perimeter above the glass; even the lowest (at top center) will only produce a light level of 10 on the grass 1 right and 3 down. Most of the lower level will then be <8 at night enabling hostiles to spawn.
Since putting something over the water sources is helpful anyway…
placing Jack o' Lanterns [JoL] immediately above the water should provide just enough light to make the corners safe. (That assumes I counted right; torches are cheap and hanging a few on the outer walls will give you some insurance.
Putting JoLs over water
Temporarily remove block to one side of water
Temporarily remove the water source
Temporarily fill water source space with any solid block eg dirt
Remove temporary dirt
Replace, till, and replant farmland
You'll also want a something atop the JoL to stop baby villagers from climbing up and jumping onto the nearby farmland.
JoLs are cheap if you have pumpkins; Sea Lanterns and Glowstone work equivalently. If you have none of these, you can use a 2 high dirt pillar with torches around the top. (Using torches like this would make adding additional torches on the outer wall necessary.)
A minor point, but the two center doors on each side will be invalid if opened as each would then have 5 transparent blocks above both "inside" & "outside".
This is unlikely as villagers generally don't leave doors open and would (at worst) limit the breeder to producing 11 additional villagers rather than the 12 Hexalobular mentions…
"Villagers will mate depending on the number of valid doors. If "willing" (see below), villagers will mate as long as the population is less than 35% of valid doors, rounded down."
(I count 40 doors, >> 14 total population possible.)
I concur with Hexalobular's opinion that wheat is a suboptimal choice; in addition to his point about seeds, wheat is a less efficient source of food.
While villager "willingness" requires 12 carrots/potatoes and only 3 bread (9 wheat equivalent), carrots/potatoes return ~2 5⁄7 per crop harvested wheat returns only 1. Thus one "willingness unit" of food requires harvesting 9 wheat but only ~4 3⁄7 carrots/potatoes.
Not relevant to villager breeders, but (as an aside) I prefer farming carrots to potatoes as carrots can be used to breed rabbits as well as pigs and need not be cooked/smelted to receive full food value. (Baked potatoes are better than carrots as a food, but I rarely eat either even early game due to the ease of chicken farming — which makes the relative food values a minor consideration.) If, however, you are hurting for sources of XP cooking/smelting potatoes will provide as much as meat or fish….
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WARNING: I have an extemely "grindy" playstyle; YMMV — if this doesn't seem fun to you, mine what you can from it & bin the rest.