For example, Razor-Qt profited from KDE and was never seen as "competition". Now it joined with LXDE and it also was never seen as competition.
Hell, even BSD profited from Linux opensource driver development by porting. Nothing by rewriting or fighting over community. Competition between opensource projects is actually very harmful.
Which healthy competition profit do you mean by mentioning "OpenOffice/LibreOffice" ?? Any proof (except worthless taskbar ofc) ? OpenOffice must be burrowed and resources should be used within LibreOffice, or he should completely change its goals to start being profitable instead of damaging behavior.
For closed source projects, them being completely different world, yes, competition MAY bring improvements in theory, but in practice it brings half-finished projects, with very short support cycle, that are quickly made obsolete by next on-purpose incompatible versions, so that customers upgrade, as in "paying again". It drains money, reassures bad quality over timescale, as well as lots of versions. This is exactly why opensource development is better and is next evolution step. If you call this "healthy competition", no further questions for you.
Second, yes, we would be a LOT better if we had ONE player, browser etc - but UNIQUE in goals. Could be 100 players with own unique approaches. But not 2 display servers targeting same goal, or two Office Suits targeting same goals. Read above - WASTE. And to add spice, its not an end solution, its a display server, its to be built upon. Having several toolkits damages the development rate exactly the same (yes, I am talking about Qt vs GTK useless battle).
The best solution was to fork Wayland and mod it, then report any changes back. In case they are needed at all. That's how stuff is done.
Gnome is destroyed thanks to Miguel and co. They abadoned user wishes, they shut the doors, then Canonical had to do something, since they never ever support or package KDE properly for unknown reasons. So they have built their own Gnome and called it Unity. Ofc it WAS different from existing DEs, so it was not met with criticism of fragmentation (except from Gnome), but with criticism of instability (Canonical's unique constant "feature"). Unity is by far not "polished, modern, productive, feature rich (you gotta be kidding me right???)" - Unity is simple and consistent in looks on all possible platforms; and thats where it ends. I worked exclusively with Unity 3-4 months, at first it was very nice, but finally it became very very limited. I got bored. Installed KDE. What a breeze. Canonical thinks now same way with migration to Qt. But this will hardly ever make Unity "feature rich".