Irreducible complexity could be falsified by demonstrating reducible complexity for the biochemical reactions cited in ID.
That is the sticking point. One side says "we don't know how they got that way...but we will someday", and the other side says "it could have been an act of creation".
It is nothing to get bothered about, both sides react to the facts but it isn't so much intelligent design that is the problem as it is the idea of irreducible complexity on a molecular level.