> the Constitution already allowed the federal government to forbid slavery
The Union wasn't (particularly the slave states in the Union weren't) fighting to abolish slavery, the Union wasn't fighting to preserve the Union, a goal to which the popular (in most of the North) cause of abolition had been deliberately subordinated in the elections prior to the war.
OTOH, by seceding, the rebel states also lost their leverage on preserving slavery within the Union, so the whole rebellion backfired. Abolition probably would have happened eventually without the war, but it wouldn't have been right away.
The Union wasn't (particularly the slave states in the Union weren't) fighting to abolish slavery, the Union wasn't fighting to preserve the Union, a goal to which the popular (in most of the North) cause of abolition had been deliberately subordinated in the elections prior to the war.
OTOH, by seceding, the rebel states also lost their leverage on preserving slavery within the Union, so the whole rebellion backfired. Abolition probably would have happened eventually without the war, but it wouldn't have been right away.