“Where am I going wrong?”
You’re not. The union’s interests are represented on both sides of the bargaining table in the case of public employee unions; on one side by negotiators delegated by the union leadership (possibly with members’votes of endorsement), and on the other side, at some meaningful level of responsiblity for allocation of public funds, public office holders that are to one extent or another beholden to unions that contributed funds to their election campaigns.
Even FDR was opposed to public unions.THAT is pretty telling.
Where you’re going wrong, is that you’re limiting your scope. Private sector unions often organize against either the taxpayer, the consumer, or both.
For instance, construction unions drive up the cost of contracts for public works — at the expense of the taxpayer.
Auto unions (along with unions in other industries) lobby government for tariffs and other trade barriers. These trade barriers allow the auto companies to pay higher wages to their unionized workforce; and pass the costs on to the consumer.