I with you that it is a problem of incentives, but I disagree that it has much to do with managers desiring more employees to increase their clout. Simpler explanations exist. If a government manager fires an employee, what does she gain? She doesn't get any of that employee's salary (the tax savings won't move the needle at all on a personal level), and won't appear to be doing a better job, since there is no profit with which to measure performance.
On the other hand this manager has a lot to lose. First, most people find putting others out of a job unpleasant. Second, that manager and her co-workers will now have to absorb the duties of the fired employee. So more work for her. Third, the department would probably be exposed to litigation by the fired employee. In my experience government agencies don't fare very well in court, and the government is scrutinized much more closely for things like nepotism and racism than the private sector.
There is another issue at work here, which is that legislators have much stronger incentives to add parts to the government than to remove them. Sure, there are plenty of voters who favor making the government smaller in general, but for any particular program the number of people who care deeply about preserving that program is always greater than the number of people who care deeply about removing it. People simply don't have the time to study all parts of the government and advocate against the existence of unnecessary but small programs. The people who are served by those programs (or sell things to the government in support of them) have all the reason in the world to care. Actual reform of the system (instead of general pouting at the size of government) would require removing a large number of small programs, and this is unlikely to happen for the reason I just mentioned.