I think this should be handled as an essay type question, rather than a math problem. Ultimately you will conclude that "there is no reason why it cannot be ergodic" but you will have a few well written paragraphs before this to review some issues and show your understanding of the subtle concepts involved.
The intuition of ergodicity is that a statistical model is ergodic if constant parameters exist (stationarity) and they can be estimated from any single (random) trajectory observed a sufficiently long time. (Reasons for non-ergodicity include: non-stationarity, or the occurrence of pathological (trapped) trajectories that do not yield valid statistical information even if observed for a long time).
As a first step, to check the stationarity, we draw the charts mentioned. The average annual returns are all over the place, but they are not inconsistent with an average return of 9% a year and a standard deviation of 17 or 18% around this value. They are very volatile, but they are mostly in a 2 std dev band around the mean, so are not necessarily non-stationary. The annual variances are more problematic, and they are markedly higher in some years (eg. 2007-2008) than others. Statistical tests (Chi Square test) can be applied to check if these are explained as sampling error around a fixed variance.
If variances are not stationary, this raises the possibility that the process is non-ergodic. However all is not lost, as there are models in which variance can switch between a higher and a lower value according to a random Markov process; if the transition probabilities and $\sigma_H,\sigma_L$ are themselves ergodic (can be estimated from a historical sample) then the process is ergodic.
In conclusion, although a simple model of constant expected return and constant variance (which would be the simplest ergodic model) can be rejected, more sophisticated stochastic volatility models (such as Markov switching between two values of vol, a lower one during economic expansions and a higher one during recessions) could restore the assumption of ergodicity.
(Which is good news, because if the stock market is not ergodic then it would make no sense to study it quantitatively).