From what I understand, the "point" of two-handed swords is that in swords, as in many weapons, "bigger is better" most of the time.

Once armor got so damn good that you no longer needed a shield (read: full plate) then everyone was happy to use the free hand for a bigger sword!

And that's pretty much just it. Would the two-handed swordsman have a special role on the battlefield? Not really. The pikemen have a special role, because a block of pikes is hard to maneuver and requires special attention to deploy. The swordsmen can simply do everything else better, so they would probably be doing everything else.

I am certain they would also be charging pike blocks from time to time. It wasn't even rare for cavalry to charge pike blocks, so why wouldn't the swordsmen? Sure, it can be avoided if you have missile troops, but when does everything go according to plan? (Never). Besides, I would think it's better to have your swordsmen charge the enemy pikes, than having your pikemen fight the enemy pikes...