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It is perfectly possible to build neural networks with other functions besides gradient descent.

I agree. I'm saying we don't know how biological neural networks are trained, which is a pretty important part of any learning construct, so we can't really know how actually similar artificial neural nets are.

What other methods are there for training a neural network? Even blind evolutionary algorithms (assuming each generation is initialized to be somewhat 'near' or 'similar' to its progenitors) are effectively descending a gradient.

To my knowledge they explore solution space but not through a gradient descent but by probing.

How can you perform a GD on non continuous function? Genetic algorithms and evolutionary methods can explore disjoint domains and functions hard from analytical pov.

Just because they can be used for gradient descend doesn't mean they use it.

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