However, they can starve to deathwith a stomach full of mature grass.
Ahhhh yes. If I can find it....I'll post a photo of some deer we saw in Huntsville, Ontario - Canada, last year......eating hay off a trailer someone had placed in their yard (near the snowmobile trails). I felt bad for the deer....for the reason you just mentioned. They could literally starve to death with a full stomach of hay.
This is why you probably see them more often eating grass in a regularly mowed lawn.
There's another reason they prefer lawns over wild grasses.