Logical Positivism didn't fall because people asked if the verification principle is verifiable; most LPists were clear that the verification principle was supposed to be analytic (it's somewhat murky what that means, but for present purposes it should suffice to note that in any version it amounts to something similar to what you suggest here). This version of history is even worse than the story that LPism failed because of the impossibility of drawing the analytic/synthetic distinction; at least that criticism was actually made, and believed by some, though it also fails to explain the disappearance of LPism as there still seem to be large numbers of philosophers who believe in analyticity or something like it. Why none of the believers in analyticity call themselves Logical Positivists any more is a complicated question, and while there are some substantial issues involved, some of it seems to involve something more like a change of fashions.
What do the philosopher call themselves then these days?
Very brief recap: The logical positivists said "All truths are experimentally testable". Their critics responded: "If that's true, how did you experimentally test it? And if it's not true, who cares?" Which is a fair criticism. Logical positivism pretty much collapsed as a philosophical position. But it seems to me that a very slight rephrasing might have saved it: "All _beliefs_ are experimentally testable". For if the critic makes the same adjustment, asking "Is that a belief, and if so -" you can interrupt him and say, "No, that's not a belief, that's a definition of what it means to say 'I believe X'."
A definition is not true or false, it is useful or not useful. Why is this definition useful? Because it allows us to distinguish between two classes of declarative statements; the ones that are actual beliefs, and the ones that have the grammatical form of beliefs but are empty of meaningful belief-content.
It seems to me, then, that both the positivists and their critics fell into the trap of confusing 'belief' and 'truth', and that carefully making this distinction might have saved positivism from considerable undeserved mockery.