Statistics are necessary for detecting reliable trends in human thought, feelings, and behaviors. Thus, they are necessary for psychology when trying understand how humans function at various times and in various contexts.
However, when seeking to help or understand an individual (either self or another), statistics may not be useful. This is because the individual is unique in a unique time and in a unique context. Statistical findings applied to a specific individual may become stereotypes.
Many psychologists have contributed to the development of statistics. So the can be said, invertedly, that "psychology is what first turned statistics in to a real science!" - let's not enter into the discussion of what is the difference between real and fake sciences.
However, the argument of humans lacking ergodicity is just a fancy re-statement of a very common idea: humanity is way too complex to be quantitatively assesed. The same can be said about several phenomena in biology, chemestry, etc. Life is, indeed, filled with complex entities that are not easily meassurable and whose behavior is not easily predicted. Yet, as a matter of fact, statistics is employed in all those fields - including psychology! The key question is whether statistics can develop better tools to evaluate such complex entities.
Personally, I am not very skilled in maths, but I am in awe at how mathematicians keep developing more intricate equations, models and theorems in order to understand complex phenomena. So yes, I do believe human beings, despite our complexity, are suitable for statistics analysis.
Statistical methods can only count the conclusions of human groups, and the conclusions of human groups can not be applied to individuals, so how to study human individuals?
Well, basic statistics gives you a (more or less accurate) map of the traits distributed among a social group. Then you meassure the individual and contrast it with the statistical map to asses where is he/she located. It does not help to explain anything, al least in this case, but to frame your descriptions of the individual studied.
Yet, the very notion of "individuality" seems to me more problematic than the utility of statistics (which of course is far from being a neutral or innocent issue). Even biologists question the concept of a "single" organisms that could be studied devoided from a set of other entities. (i.e. "we have never been individuals" Article A Symbiotic View of Life: We Have Never Been Individuals