I don't remember where but I saw a chazal that states something to the effect of "the (non-Jewish) leaders of the world do not have free will". I would appreciate someone editing to source this. So I am wondering A) Does the President have free will and B) If not then does it matter who we vote for? (Even if we have a chiyuv of hishtadlus, I am asking if the choice will ultimately be effective) I would like to stay away from the fundamental discussions of bechira that underlay my questions and stick with this particular point. Thanks!
The famous saying that you are thinking of is לב שרים ביד השם. Unfortunately this saying is not from Tanach, rather it is an adaption of the Posuk in Mishlei 21:1 לב מלך ביד השם.
This is generally understood that leaders do not have free choice on things that affect their subjects, rather God decides that himself and sways the Kings heart. For more of a discussion of this and about King Pharoah, see this Aish article which discusses this at length.
As for whether the President has free will, it follows that on things like Iran which would affect millions of citizens, he does not have free will. The Aish article seems to imply that by doing the right Hishtadlus and voting for the right person, God sways that leaders heart to the Good!