By this, I mean the assumption that there is some universal logical structure to human language, and that our job is simply to come up with the labels for each constituent part and arrange them all correctly like a jigsaw puzzle. Once the puzzle is complete - boom! We've "solved" language.
That would have been a reasonable starting point back in the 60s, but I don't think it's advanced the field nearly as far as statistical methods.
Chomsky would retort that "advancing the field" is meaningless unless it allows us to "understand the field". But if natural language is a largely arbitrary collection of rules in constant flux, there would be very little of this "understanding" to be had. You could be bashing your head against the wall for centuries trying to induce structure into a bunch of symbols where none was ever to be had.
Now I do believe that purely statistical methods will hit a wall - the same way you could never teach a baby to communicate by throwing it a copy of Wikipedia and nothing else.
But if some parts of language are essentially arbitrary, then statistical methods are the best available tool to uncover that.