While public unions have their cons (like private unions) given the size of the public workforce, one can easily imagine a worse result in the absence of public unions.
Ignoring postal workers and teachers for the moment, one of the things that unions do is formalize the process for vacancy filling and vacancy creation (this being only one example). This force does much to disrupt cronyism, regardless if you experience specific examples to the contrary.
Public Unions interfere with the ability of an elected official to pick a few department heads, those department heads to pick their closest family members as deputies, the deputies picking their friends, and everybody right down to the janitor essentially being a political appointee. An election brings a swap-out, and after a few cycles one is left with nothing but ineptitude.
Unions bring process, improve transparency, and bring a method for systematic abuse to end up in the courts. They create their own problems for sure, but I fear we lack a decent negative control for what the state of affairs would be today had we not had public sector unions. I suspect the relationship between effective government and unionized government is perhaps the opposite of what you suggest. The Social Security Administration and the Internal Revenue Service (e.g.) are two agencies that regardless of the perception in the public and through the press, are remarkably well run and remarkable efficient. My personal experience with e.g. the FDA has consistently been one of very capable individuals with large workloads capably managed and remarkably ethical given their power. The overhead versus throughput of these agencies is something the private sector can only envy, to the extent that they are inefficient, the blame lies in plenty of places, but not at the door of Union representation.
Russia and China don't have public unions, and my understanding is their governments are best understood through an understanding of genetics and zip codes.
But I could be wrong. And certainly I have experience with agencies staffed and/or run poorly. But you draw a line connecting taxes-accountability-effectiveness that seems to pretend that somehow one can create large institutions that don't somehow suck, that getting rid or public unions would somehow make them suck less, or that by virtue of them being funded by taxes that one is somehow reasonable in expecting an institution devoid of waste, abuse and suckiness. Until such a thing occurs, even just once anywhere in the known universe, I am not sure that the taxpayer criteria is a reasonable basis to require that which has never existed anywhere (despite constant effort and desire) to suddenly exist here and now.