When did mythology replace historical fact?
A small but very vocal segment of the nation (USA=both North and South) opposed the war for various reasons, that does not mean the war had no support in the North or that the Union resorted to the draft because it could not fulfill its enlistment quotas.
A few points,
Enlistments overwhelmed the ability of the system to provide for incoming soldiers in the first year of the war, in fact many regiments that served in the Western theater were not fully outfitted and equipped until 1863.
Both armies expanded at an incredible pace in the first 20 months. Equipment, uniforms, weapons, horses, facilities and command personnel were not in place to handle the demands. The draft was a way to schedule inductions.