Aside from what has been covered regarding habitat and browse quality, I wonder if deer in proximity to cattle do not benefit from crossover of gut bacteria. Both are ruminants. A quick google says cattle are more effective at digesting grasses and cellulose.
I remember a situation from Australia where the camels, once considered a pest on the western cattle lands, were found to actually benefit the cattle two ways. First, the camels ate weeds that competed with the grasses. Secondly, the cattle that shared water sources with camels were fatter than those which did not. This was due to the introduction of camel gut bacteria’s which more effectively broke down their forage.