Personally, following a vegan, or even doctor recommended diet for diabetes I gained weight on both... on keto, I'm down 63# last year, and feeling much better and mostly off medication. I've also got a dysfunctional metabolism, so YMMV.

Further, it's absurd to focus on the macro-nutrient ratios of a single calorie source. For most people, dairy is almost invariably balanced against other calorific sources, many of which are almost entirely carbohydrate, like wheat, rice and refined sugar.