Supplements/fortifications are a good idea for any kind of diet especially nursing mothers. No question people who eat a vegan diet are susceptible to deficiencies in B12 (and they have to eat supplements or food fortified with B12) and to a lesser extent calcium and iron (very diet dependent). Omni's and Veggie diets are both going to need supplements in Vitamin D (unless one live in a very sunny local) and Iodine (most people get this from fortified salt).
That being said B12 is not an animal based nutrient, it's bacteria (in the soil) based. Omnivores get it from animals they consume because the animals got it from eating unwashed plants that have the B12 producing bacteria. We now live in a society where food is thoroughly washed or grown in "sterile" soil so we can't get B12 from eating root veggies.
We clearly are omni (extremely good survival mechanism) but I would argue the historical omni portion of humans diets are considerably less than our modern day diet. Chimp's meat/insect portion of their diets only account for about 2% of their calories, rest is plant based.
As far as the parents in the OP, obviously completely ignorant as are parents like these: