i think its relevant that the liar has to say false things, which is more limited than just not-true things.

if you dont assume the vacuous truth, and instead leave it undefined, then when hes got no hats, "all my hats are green" is absurd, rather than false.

the gotcha only stops applying when you put a vacuous false, rather than true or undefined.

is this really a flaw in applying classical logic? with the vacuous true, the only information you get from "all my hats are green" being false is that they have at least a hat, same as the intuitive result