You're forgetting about the implied sub-selection (or lack thereof), which then needs to be re-applied to the situation over which the group has power.
Take random small town A containing subgroup B, which holds most seats of power in town A. This is the same as: Subgroup B, as a class, having power, within town A.
So let's rephrase: American men, as a class, have power in America.
Now, what happens if our sub-selection ISN'T a sub-selection, and is just "humanity"? Human men, as a class, have power in humanity.
This is backed up by leadership statistics.
Now, I think what you're trying to say is: Correlation of the "man" attribute with power is the not the same as the "man" attribute causing power".
Again, this kinda makes sense, but, AFAIK/IMHO, so long as the attribute in question is disproportionately shared by the people with power vs the distribution amongst the population, that attribute IS causing some degree of the power held.