Is ever a covariance stationary process, in the real world, not mean ergodic? Mean ergodicity for a covariance stationary process is a property that assures that the sample time mean $\bar{Y}=\frac{1}{T} \sum_{t=1}^T Y_t$ converges as $T \rightarrow \infty$ to the ensemble mean $E[Y_t]=\lim_{N \rightarrow \infty}\frac{1}{N} \sum_{n=1}^N (Y_t)_n$ (ie the sample time mean is consistent).
For the sample time mean not to converge, the process needs to completely "forget" about its ensemble average as time passes, so at a certain point adding new observations to the sample mean won't add new information. This should imply that the process can't infinitely "gravitate" around its ensemble mean. Please correct me if I'm wrong about this.
This is clear, as an example, for a random walk process; even if the process starts at $Y_0=0$, the new value $Y_{1}$ will "reset" the mean for the following observed sequence of values, ie the following part of the time sample completely forgets about the starting 0. However this process is not stationary, as its variance increases with time.
Going back to covariance stationary processes: any process that is covariance-stationary is not necessarily mean ergodic.
Now, the process can't "gravitate" around its ensemble mean, but It also can't progressively wander off as a random walk does. How would it behave?
The examples I've seen are covariance stationary processes of this kind: $Y_t=U+ \epsilon_t$, with $\epsilon_t$ white noise. Here $E[Y_t]=E[U]$. The sample time mean won't converge to $E[U]$ for a given sample $\{u_i+e_1, u_i+e_2, ...\}$ as it will converge to the single realization of $U$ for that given sample ($u_i$).
So the behavior of this kind of processes is summed up as: the process gravitates around some value which however is not the ensemble mean of the process. Are there any covariance stationary, non mean ergodic processes that don't exhibit this kind behaviour? The sample time mean has to converge to something for each time sample, given the variance of the process is finite, right? (it just can't converge to the ensemble mean)
Now to the final question: assuming we will ever get to experience only one realization (path) of a process (so excluding situations where you can look at multiple paths), then $U$ "doesn't exist by our point of view". Our reality is $u_i$, and the process will be ergodic for estimating $u_i$, so we wouldn't know the process is not truly ergodic (by looking at multiple paths), and we wouldn't care anyway (since we are trying to model and predict the future of our realization, our reality, we don't care about $E[U]$). In this situation, a process not truly mean ergodic wouldn't be a real problem.
Now, a covariance stationary process, of which we can't even consistently estimate a constant value (like $u_i$) as the mean of the single path which is our perspective, would be a problem.
Does a covariance stationary process like this exist?
(this is what happens with a random walk, but the process is not covariance stationary)

As you point out, at least for "real" processes which happen over historical time, there is no way we can tell whether they look ergodic or not: a single realization is all we get to see.
I think is was A.J. Ayer who said that, for a question to be meaningful, we should be able to describe an experiment whose outcome would provide an answer. Inasmuch as we cannot experiment reversing time, we probably cannot answer your question for time processes.
I conjecture that we could obtain empirical results close to ergodicity for processes not indexed over time. For instance, if we could measure a magnitude (e.g., temperature) along infinite paths in different directions starting from planet Earth, it is conceivable that the averages measured along each "ray" might be close to the ensemble average. But this is mere speculation.
