The only problem the author points out is that he doesn't like the Cantril Ladder question.
I get it if you feel like that question falls short of representing your own personal concept of happiness, but that question is the standard in positive psychology research for measuring self reported subjective well being, and hardly enough to say the report is "beset with methodological problems".
The more practical problem is that the samples used in the Gallup World Poll are for largely unavoidable reasons small and not representative of entire country demographics; in particular respondents can skew richer and more educated than their national average in poorer countries.
They give several well-considered criticisms of the question - it leads people to focus on socioecomonic status, it doesn't correlate with other measure like whether they report experiencing joy recently, etc. It's not much of a defense to simply say "well, it's the standard".