That used to be exactly the case. They changed the laws since then since each president would fire everyone and replace them with his own people. Now you have to be hired for merit and can't be fired for political affiliation. This is the "Civil Service".

Restructuring an agency(and removing the positions) is different from replacing people, though. I'm sure there's something that could be done.

Sure that's what they tell us. In fact the spoils system served an important purpose: it limited the size of the executive branch, since you can only hire so many people at once. So much for that.