And zombies would be no kind of predator for anything but the most sickly deer, don't be stupid. The stamina argument works based on dogged pursuit by intelligent beings that can track them. A deer running for 20 seconds would be basically vanished to whatever zombie initially went after it.
Don't overestimate how much distance a deer can clear before giving up, it's not that much.
A deer running at top speed for 20 seconds will clear around 800 feet in an open field. It isn't going to be able to sustain that speed for more than a few 10 second bursts though before becoming exhausted. That is far enough to break line of sight with a single zombie most likely, but we aren't talking about a single zombie.
Herds in Walking Dead have over 2,000 zombies in them frequently, a group of that size will have an effective radius of detection of at least a couple thousand feet in diameter. A herd would most likely kill any large game it came across.
You don't need to be a master tracker to keep up with something that can only move in short bursts of a few hundred meters when you have an unrelenting search party the size of a Walmart parking lot. It's not like the zombies will lose interest, and they are fairly cohesive in large groups. If one on the perimeter picks up on the deer a good portion of the group will continue to move in the general direction of the deer until it is flushed out again. Deer aren't smart enough to stay quiet in hiding under brush as thousands of zombies pass it, if you get close enough it is going to make a noise, if it is exhausted it will be constantly making noise.
Not to mention, areas where animals will tend to congregate, such as water sources and good pastures, will have a higher density of zombies simply because they are constantly lured there by animals.
In an area densely populated with zombies, a deer won't get very much reprieve.