That does not fit the evidence.
And besides you can read thousands of articles on HN about anxiety in humans, a mostly useless anxiety focused on societal 'threats' which we suffer from just as much.
At least a deer is on the lookout for something real.
Also if you compare animals lives to human ones, with our propensity for war and torture and persecution, I think the animals _do_ objectively live calmer lives.
You don't see them systematically tearing each other to pieces over made up goods like money.
I think this trope that "nature is a constant struggle" is a projection of human values (or lack of) onto nature.