I think you're right. The brain is a very specialized computing machine, and it evolved for a specific purpose.* Evolution doesn't reward general computing capabilities. Hence, computers are vastly more general than brains. If something is, in principle, uncomputable, than there is no chance that a brain can compute it.
*Sure, a networked system of neuron-like "units" might have general computing capabilities. It's as if you saw a fully functional modern computer for the first time, and it only gets fed arithmetic problems - the system itself is much more powerful and general than that, and you could hack around and make it do other things once you understand how it manages to do arithmetic. The brain is kind of like that too -- it gets fed stimuli and it has evolved to process those stimuli. Yet it might be that a computational system made of a network of neurons could be a lot more powerful and general than that in principle.
*Sure, a networked system of neuron-like "units" might have general computing capabilities. It's as if you saw a fully functional modern computer for the first time, and it only gets fed arithmetic problems - the system itself is much more powerful and general than that, and you could hack around and make it do other things once you understand how it manages to do arithmetic. The brain is kind of like that too -- it gets fed stimuli and it has evolved to process those stimuli. Yet it might be that a computational system made of a network of neurons could be a lot more powerful and general than that in principle.