This is not about the example. What I'm saying that
no betting at all should care about the time average. Betting is about having good estimation of
outcomes, and time averages only helps you when the process is ergotic.
That's a very special case. For everything else (that is, non-ergotic processes), your time average is crap, and you must look at the distribution of outcomes directly. Even the ensemble average is not enough. Averages are crap at visualising skewed distributions. For those you want the median, the quartiles, sometimes even the percentiles.
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To be honest, this "ergotic theory" shows signs of snake oil. The definition of ergodicity itself is dead simple, so it's pretty easy to evaluate. What seems pretty clear is that ergodic processes are the exception. And a pretty uninteresting one at that, since it's a class of processes that people will have good intuitions about.
It would then seem that ergodic theory is more interested in the non ergodic processes (the very point of this blog post is to warn us about them). That is, processes that lack some property —the general case. And surprise, since the time average and ensemble averages are different, and you only care about the ensemble average (well, the ensemble distribution really), the time average won't help you. Be afraid, or lose your assets.
That's why I see snake oil: what works on non-ergodic processes will also work on the ergodic ones. Unless you need to make a split second decision using your intuition (which while inadvisable is safer with ergodic processes), there's no need to make the distinction at all. Just analyse your process without without assuming it will be ergodic, the results will be applicable even if it is.