I mostly agree with you, but I want to add that GNU was the leader from the start with the aim to create a complete, integrated operating system, rather than just a bunch of unrelated programs tossed together. It was not important to them that all the code was written by GNU, more so that there was a complete free system.

The idea was that one project worked on the display server, another on the desktop environment and so on, with the intent that all come together as "GNU".

And then Linux came and took the name of what GNU anticipated to become.