The author seems to miss that "economics" is a vast field spanning the whole spectrum from applied economics, over theoretical economics, mathematical finance, financial mathematics, to pure mathematics (with physicists working along the whole spectrum, so this is not a consequence of the author's background per se).
He fails to engage at the right level. From a theoretical mathematician's point of view all models are wrong - they are just deductions from assumptions. From an applied economist's point of view are models are right - they explain some observed phenomena.
Ergodicity is not a niche topic, most intermediate courses on stochastic processes will cover it. Will loosening an assumption about the properties of stochastic processes yield different, potentially better models? Maybe. Will it lead to a revolution in economic theory? Unlikely.
Again, odd that this has been peer-reviewed.
See also the reply here, which is rather damning [1]
And the author's reply to the reply. [2]