I would imagine that the antibiotics in cattle issue is far more closely related to factory farming in confined quarters than anything to do with what kind of food they are primarily eating.
There's a class of antibiotics (ionophores) that are used in cattle feed specifically to manage issues with grain feeding. They reduce incidence of "bloat", and increase feed efficiency. They are also used with other feeds, but the purpose is still to manage digestion issues.
I think there is more use of other antibiotics to treat sickness in feedlots though.
The two are not mutually exclusive; each carries a significant tax on the animal's health and immune system.
Humans are fairly unique in our dietary flexibility; most other animals are evolutionarily adapted to a narrow range of foods. Corn provides caloric energy, but little more; a cow needs to eat grass, and move around, in order to be healthy.
