The GPL is not simply about fixing bugs. It's about preventing the existence of versions of the program in which you cannot easily find or fix bugs, and the underlying ideology that they should be no such programs. And not only bugs but deliberate undesirable or malicious behavior. The problem with closed source proprietary software is that you don't know what's hiding in the binaries.
If software were uncopyrightable, they would still be binaries without source code, which you would have to reverse engineer to find out what harm they perpetrate actively or possibly through their security flaws.
Moreover, if you wrote a piece of free source code, anyone could do anything with it they wish, including removing your name, and not attributing you in any way in the documentation accompanying the compiled code.
Stallman was a control freak who insisted that people modifying his code give back the contributions to the project. In the world without copyrighted programs he would have had no leg to stand on, and he knew that. A world in which programs are not copyrighted would need laws which ban the distribution of binaries without buildable source code, and all the tools needed to build it and their source code.
A program is not distributed to you if someone else installed it on a machine you don't own and you're just borrowing that machine. It doesn't matter if you're physically at the console, we're at a remote dumb terminal or smart client. The program was distributed to your school, company or friend or whoever.
I don't think that even the people who drafted the AGPL believe the nonsense doctrine that the visitors to a server have been distributed the software. That's just something invented by the downstream AGPL apologists. (And of course vendors of proprietary software like the doctrine also. If you buy a program and let 10 people use it via remote access to your machine, they would like the legal system to believe that those people were distributed the program and that you should buy ten more licenses).
The doctrine is not required for the AGPL to work. The software distributed to the individual or organization installing it and running it for visitors. The license is concerned with the behavior of that individual or organization, and uses the power of copyright to make their permission to have a copy of the program conditional on their usage behavior. The visitors to the running application are not parties to the license.
The GNU AGPL contains these definitions:
> To "propagate" a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.
> To "convey" a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.