I've never met anyone that only eats meat. They may eat a meat-heavy diet, but I've never seen anyone exclude plants altogether.
I am a vegan, and my person expectation is that excluding animal products altogether is not the optimal diet. The optimal diet, I believe, is still primarily plant-based, but would include modest amounts of meat (mostly fish). Dairy would probably be excluded altogether, as milk is not something that animals consume once they are fully developed into adults. That said, a modern vegan diet is close enough to still be healthy.
I don't exclude animal products from my diet because I think it is the healthiest. Rather, I exclude them on the basis of moral/ethical grounds. The slight health benefit that adding modest amounts of animal products to my diet would provide are not worth the blood I would have on my hands.
As for missing nutrients, the only one that requires particular attention is B12. I purchase a 1-year supply of B12 supplements for less than $5. It should be noted that even people who eat meat (omnivores), the animals that they are eating also have been given B12 supplements. Either way, you are getting B12 through supplements.