I think he doesn't describe GCVOE but rather uses it to explain the apparent paradox.

Naively expecting "better" organisms to win fails to explain why these birds are evolving progressively "worse".

But if you go down and look at genes which handicap the male while triggering some silly attractiveness glitch in females versus genes that don't, it becomes clear that as long as the handicap isn't too strong, the former genes may easily win.

I think that's a point for the GCVOE this time.