I would agree that animals do not spend their lives carrying messages around for humans. I prefer to focus on communication: when you come into contact with a wild being (and it could also be a tree or a stone) something is communicated by the encounter. It follows that you can make your own meaning out of it. If the dear "spoke" to you, what did the deer say, i.e. what did you perceive from your encounter with the deer and what does it mean to you?
Sometimes, I do believe that animals communicate with us in a more direct way. A friend told me that once, when he was snowshoeing alone in the Alps, a chamois he hadn't seen startled him by "barking" at him (it sounds like a raw sort of barking, I'm not sure what it's called). He took it to mean that there was danger ahead and he shouldn't continue - he felt that very strongly, and turned around and went home.
In most cases however, I think it's mainly about us not being used to contact with the wild. Animals live everywhere around us whether we see them or not, and I would much rather work to reestablish communication with them rather than turn them into symbols. There is sacredness in that attitude also.
Hedge-bandit, song-bomb, dart-beak, the wren
hops in the thicket, flirt-eye; shy, brave,
grubbing, winter's scamp, but more than itself–
ten requisite grams of the world's weight.
—Carol Ann Duffy