You didn't answer the question. Why would you want to learn anything from deer?
Why would we want to learn anything "from the lillies of the field or the birds of the sky"? Let me answer the rethorical questions.
"Why did Solomon, David, Job and even Our Blessed Lord want us to learn anything from animals?
Why did God create deer anyhow? Exclusively for meat and hide?"
When God created deer or any other animal, he was looking at himself as the exemplar, the Catechism of Trent says this:
As it was His own goodness that influenced Him when He did all things whatsoever He would, so in the work of creation He followed no external form or model; but contemplating, and as it were imitating, the universal model contained in the divine intelligence, the supreme Architect, with infinite wisdom and power-attributes peculiar to the Divinity -- created all things in the beginning.
So each animal is a reflection of a certain aspect of God, and in turn of Christ and of man.
We live in a materialist world that confuses the "how" for the "why"
Why do birds sing? Most people would say "to attract mates, define territory, and warn of danger" and that explains the material causes, but the deeper reason why birds sing is closer to something like "to show us that we should also constantly happily sing the praises of God, and that Christ sings with joy through his body The Church."
And you could say "the Bible already instructs us to do that we don't need to ponder about birds" but I don't know I feel like when God made it so that everytime you go out you hear and see birds it wasn't "just because" and we should just ignore it. The people of Christendom certainly didn't.