I'm not a milk specialist but it honestly seems to me a typical type excuse that a farmer or supplier might say just to keep a customer or even to rationalize something to themselves. There is a pretty amorphous range of 'grass-fed'/finished and exactly what constitutes grass, but I don't think everyone should be concerned about their beef necessarily for this reason as I don't think its as widespread as your source says.
I had a seller at a farmers market have the audacity to tell me that all scallops were harvested and sold frozen, and there was nothing they could do about it, even tho every StopandShop on the coast sells them fresh from the boat.
I've only been to a few farms, but I have never seen or heard of evidence of the above. The one farm here I can actually get milk feeds the cows no grass or grains. its like 100% barn hay. I would probably prefer mostly pasture and a little corn grass myself, as I don't find what I can get very healthy.