The Taliban were willing (after a bit of threatening and cajoling) to hand Osama bin Laden over to a third party for him to go to trial. There was no need to invade based on the justification as the Taliban were genuinely afraid of the invasion and were willing to co-operate, just as they have been now. In the end, the invasion did nothing anyway and Al Qaeda's peak came AFTER the Taliban was toppled. There was never any chance of a cohesive post-Taliban government emerging from the Northern Alliance. By this point the US had decided on war and the whole MIC machinery was rolling, so it was too late to turn back (as US leaders thought, with their reliance on a captured media and lobbying from the MIC creating strategic liabilities within the US state).

The invasion was not necessary for US security aims and certainly could never have bettered Afghanistan, though.