What makes the tale so “fresh” is that the author of the book the movie is based on made up the whole story. Lincoln and Grant were busy running the war during those months. Republican abolitionists in Congress pushed through the 13th Amendment after his death as part of the reconstruction penalties on the defeated South.
The 13th amendment doesn’t say anything about all men being equal. It prohibits slavery, a different issue. Lincoln, as President, had no Constitutional role in getting a Constitutional amendment passed by Congress and sent to the states. It was adopted by Congress on Jan. 31, 1865, before the end of his first term, but was not ratified until Dec. 6, 1865.