Where did generative semantics go wrong? Why was their conception of language faulty?

I think "going wrong" is probably not a good way to characterize the development of Generative Semantics. What happened is that it's various proponents went in different directions - neither Ross, McCawley nor Postal were interested in pursuing the programmatic work and spent the rest of their careers on more wide ranging contributions but ultimately partial to the study of language.

George Lakoff never really abandoned the Generative Semantic program of putting meaning at the center of linguistic inquiry but left the generative, formal linguistic approach. His assessment of Generative Semantics (which can be found most clearly in 'Women, Fire and Dangerous Things') is that it was making unwarranted assumptions about meaning as being subject to formal mechanisms contained in mathematical logic - which Lakoff calls 'objectivism'. Lakoff rejected that in favor of an enriched version of 'frame semantics' having also jettisoned formal syntax in favor of construction/cognitive grammar. So you can say that generative semantics simply evolved into 'cognitive grammar'.

If you look at some of Lakoff's writing from the early 1970s (e.g. hedges), you can see a clear continuity of his overall program in the things he's concerned with.