I agree with your definition of ergodicity and performativity. I want to take the argument a little further. As I understand Peters, money/leverage is a device to get access to parallel results. You can transform the example into a ergodic process if you only bet a small fraction of your wealth/budget. In neoclassical theory it is ruled out that you go bankrupt because a favorable outcome will always be funded. This is functionally the same thing as leveraging(?) and guarantees the market efficiency in neoclassical theory.
Nassim Taleb also has an interesting post on ergodicity: https://medium.com/incerto/inequality-and-skin-in-the-game-d8f00bc0cb46?source=rss-f138bf5466fe------2