There is a much less paranoid reason that an organization like the FSF might want copyright assignment. Suppose I take some portions of a GNU project program and incorporate them into my proprietary program. The FSF takes umbrage at this, and sues me for copyright infringement.
Without copyright assignment, I'm going to go over every single piece of code they say I took and I'm going to allege that the FSF does not have standing to sue because they are not the copyright owner for that particular piece of code. They will have to prove that the pieces of code they are suing over are just ones that they have the copyright on. The accuracy of how they track code ownership will become an issue, and I'll get to annoy their admins and IT guys with depositions about all that. It makes the lawsuit much messier and annoying to the FSF. Or maybe they can add all the contributors as plaintiffs, if they can get them to agree, and that would blunt my ownership arguments, but now I get to hassle all the contributors with depositions!
If the FSF owns the copyright on the whole program, all of that crap goes away. They just file copies of the copyright assignments with their lawsuit. If I want to seriously question their right to sue over some of the code after that, I'm going to have to find a way to argue that the assignment was invalid, such as arguing that the contributor did not have the right to assign copyright. That is going to take a lot of effort on my part (or on my bank account's part to pay investigators), and if the FSF was reasonably careful it is unlikely to pay off for me.
In summary, if you take code from other people AND you intend to legally try to enforce your license, then you probably should get copyright assignments. If you don't intend to enforce the license, then there is much less need.
The FSF is at least as much a political organization as it is a technical organization, and to achieve their political goals they need to enforce GPL, so they need copyright assignment.
If I, on the other hand, were to release a project under GPL, I would not care at all about enforcing GPL. I normally use a BSD license, and the only likely reason that I would be releasing a project under GPL is that I wanted to use someone else's GPL code in my project and so used that license to allow this. Hence, there would be no need for me to ask for copyright assignment.