You are ignoring the fact that it was a Constitutional amendment, not a bill. And that as president there was nothing that Lincoln could do to further it or hinder it or impact it in any way.
You don't think a President can have any impact on a constitutional amendment by coming out in favor of it?
So what then of Lincoln's pushing of the 13th amendment? Was all his wheedling, bribing and threatening of no effect on that amendment?
I thought there was a recent movie about Lincoln, the entire premise of which is that the President got the 13th amendment passed with tricks, bribes and threats, despite all opposition. (And a very near thing it was too, according to the movie.)