I hate spoiling everyone's fun, but the deer does want to be put down.
When the guy is holding the deer upside-down, it instinctively goes rigid. When he starts to put it down, the "escape" instincts kicks in and it starts struggling & yelling for mama.
The little deer DID get back to mama, though. And it is good to see they saved it rather than just shrug their shoulders and fell a tree on top of it.