Am I right to think that this statement is based on the assumption that the brain (and all computation machines) have been proven to have Turing machine equivalents based on the Church-Turing thesis?

No, otherwise we would have the certainty that the brain is Turing-equivalent and I wouldn't have prefaced with "Either the brain is capable of doing things that would require infinite resources for a computer to perform".
We do not have proof that everything not calculable by a Turing machine requires infinite resources, otherwise Church-Turing will be a theorem and not a thesis, but we have strong hints: every hypercomputation model is based on accessing some infinite resource (whether it's infinite time or infinite energy or infinite precision). Plus recently we had this theorem: any function on the naturals is computable by some machine in some non-standard time.

So either the brain can compute things that a computer would take infinite resources to do, or the brain is *at most* as powerful as a Turing machine.

As per the electron thing, there's a level where there is symbolic manipulation and a level where there isn't. I don't understand why it's symbolic manipulation for electronics but not for neurons. At the right abstraction level, neurons too manipulate symbols.

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