Animals like us have plenty of control theory built into our firmware, even if we aren't going through the calculations on the cognitive software level.
A control system, tuned and running, is a few lines of code in a fast loop, or a feedback amplifier circuit. Just like a neural net, it takes a while to tune the coefficients but then it becomes muscle memory. And since much of control theory is fully general (like Bode's sensitivity integral), those principles must also hold for biological systems as well.
A baby doesn't think about Laplace transforms when learning to walk, but neither do they think about backpropagation.
If you're keen to replace control theory with something else, I recommend solidly understanding it first.
That said, deep learning is great for when we can't model things well. For example, there are a number of mathematical issues with how we model object grasping even before visual data is introduced. Since the setting is so difficult to work with analytically, there have been a number of exciting data-driven solutions proposed in this space.