They have wielded influence but there's a major difference between those two parties. The republicans are way more hawkish and won't hesitate to use military force while the democratic agenda is based off less war and more economic development. Obama just plays them, after he got his second term he doesn't care what they think, he wants to open a new chapter between the US and Iran. What's stopping all this potential progress however is the congress which decides what we implement, so that's the problem and that's where AIPAC does most of its work. If Obama had congressional support a deal would have been reached by the end of his term. He can't get that because of republican opposition which seeks to demonize him in every way.
I don't see him getting the position, he doesn't have the prowess of a powerful president in him.
I say Howard Dean would be my choice and I wonder what he can bring to the table.