Being in the cattle business and having taken animal science/range science while in college, I have a little knowlege regarding this subject, albeit limited and far from an expert.

One of the first things you learn in regards to managing native pasture/rangeland is that cattle, sheep, and deer/goats are complimentary to one another in maintaining a proper balance of range plants in a properly managed situation in that their PRIMARY food source is different from the other species.

Native pasture plants can be broken down into these general categories- grass, forbes(weeds), browse(woody plants and leaves) and legumes(clover type plants). Legumes, in a native situation comprise the smallest percent as they are usually the most preferred by all species.

However, animals, just like us like a variety in their diets.
While these aren't exact percentages (I am relying on memory from 30 years ago) these are close enough to get the big picture of what the different species consume in regard to variety. I am sure I can dig through my textbooks and get the exact if someone pressed me.

Cattle- grass 70% forbes 20%, browse 10%
Sheep grass 30%, forbes 40%, browse 30%
deer/goats grass 10%, forbes 10%, browse 80%

So yes deer eat grass just like cattle eat browse, but the percentages of eachare on the opposite end of the spectrum.

It will also vary within times of the year and what else is available that is nutritous. Wheat and oats are classified as a grass yet is the preferred food source for deer during winter months.

Alfala is a legume. Peanuts are a legume.

As far as deer starving to death on hay, so can cattle if it is too low in protein and TDN (total digestiable nutrients).

There is a saying"A chain is onlyas strong as its weakest link." This statement is never more true than when it comes to Nutrition, regardless of the species in question.