Since brain has to work in a noisy environment, it is unlikely its function depends too much on details. Neurons can't depend on effects we don't know about, because if effect is large, we would know about it, and if effect is small, brain has to work with noise that size, so it is not necessary for us to get it right for its function. (It's a different matter for its exact output, but that's for later.)

You don't need to appeal to "quantum effect" to think that being able to understand, well enough to simulate, THE SIMPLEST NEURAL SYSTEM of any organism would be helpful before jumping in feet first with 1 billion euros into the most complex neural system. Of course it is a process over 10 years and they have "only" spent 200 million so far. I would expect that quite a bit of that money is going to study model organisms from c. elegans to mouse to chimpanzee. Studying simpler model organisms is one of the best ways we have to understand fundamental biological principles.

If there would be such a quantum effect it would contradict a lot of conventional theoretical physics, brains operate at high temperatures ~310 Kelvin, and the length scales involved are fairly large for quantum mechanics. You would only expect to see quantum effects at such large length scales at very low temperatures, because otherwise the correlation length is just too small.