This would solve the problem of aggregating complaints. The first line of defense of a large entity is information asymmetry: they isolate the customer, make them feel like their problem is unique, or that it's no big deal, no-one else cares, etc. In this case, United (correctly) realized that PublikDemand's twitter account was successfully serving as a de facto complaint database, and acted to remove it.
Knowledge of other customer's complaints is a powerful weapon for any customer with a problem. It tells them that they are not alone, that the company has a problem, and opens the possibility of combining forces to take action against the company: the cost-burden of legal action against the company can be shared between plaintiffs, a kind of psuedo class action suit.
As it stands, only the most sensational horror stories are seen, and they are not reported in a way that is easily searchable or usable for the next customer that is abused and whose abuse is buried by information asymmetry.