You can’t generalize about the entire public sector from a couple anecdotes. Of course there are going to be badly run groups inside an organization of that size. I can tell you there exists management inside the federal government that runs performance management and fires underperformers.
Since you fall victim to such an obvious fallacy, it makes sense that you would wind up hired into low-performing groups.
Trust me, basic performance management processes that exist in the private sector are almost entirely absent in the public sector.