Currently I am raising 2 Jersey heifers (3-1/2 & 4 months), they have not yet had a bite of grain -- but they are still on milk, graze with the cow, get 2 lbs of alflalfa pellets daily, and free access to grass hay.
Over the years that I have raised both beef and dairy calves, what I have found is that early weaned calves need grain/supplementation to grow properly and not become potbellied, later weaned calves do just fine on forage alone.
But the biggest problem that I have found in raising heifers is that over feeding of grain causes them to deposit fat into the udder, which displaces mammary tissue, resulting in lowered milk yields over the animal's life (once that fat is in the udder, it's there forever). There is also some evidence that overfeeding grain also causes them to deposit fat around the reproductive tract, making it harder for them to conceive.
Dairy cattle deposit fat internally before they deposit fat externally.
Animals that are raised on grain tend to need grain just to maintain condition, while animals that are raised on a forage based diet tend to be much more efficient at maintaining condition on just forage, and only need grain when milking heavily.
Of course, genetics plays a large role in milk production -- you can't take a heifer with poor milk genetics and turn her into a powerhouse milker by raising her on forage. What you can do is maximize her milking potential by feeding her as Nature intended a cow to be fed (forage based).
Dairy cow and dairy goats are quite similar in their needs and performance -- during the years when grain was rationed during WWII and growing does weren't getting grain, milk production increased dramatically. Some of those WWII milk records still to this day, haven't been broken.
Ann B