Even in supervised ML, pure gradient descent is not the most efficient optimization strategy. E.g., momentum is ubiquitous, and the updates it induces cannot be expressed as a gradient of some scalar loss. But the rotational non-gradient component of its updates substantially improves performance and convergence on the architectures we use.

The brain probably primarily uses something like TD for task learning, which is also not expressible as a gradient of any objective function. And, though the paper mentions Hebbian learning, it's only very particular network architectures (e.g. single neuron; symmetric connections) that you can treat its updates as a gradient of some energy function; these architectures aren't anything close to what we see in the brain.