I think the intent behind the modern usage of "the exception that proves the rule" is that:
a) the rule is a heuristic, not a hard-and-fast rule, and
b) the notability of exceptions suggests that the heuristic is pretty good.
In other words, if the exception is, in fact, exceptional, it reinforces that the heuristic is reasonably valid for non-exceptional cases.
In this case, the fact that it took the intervention of the family-friend-who-works-at-Google-HQ to resolve a pay dispute is an exceptional circumstance, and reinforces that for most people, because presumably most people don't have a family friend who works at Google HQ, there is no recourse to pay disputes.
Hence, the exception—by being exceptional—proves the heuristic rule that most employees of these contractor corps have little power to negotiate for their rights.