How did emancipation make military sense?

This was a wartime tactic that ensured the Union could draw on the manpower of thousands of free Black Americans.

Emancipation during the Civil War made military sense for the Union for several reasons. Firstly, it helped to deprive the Confederacy of valuable resources by freeing enslaved individuals who were contributing to the Southern war effort. Emancipation also served as a way to undermine the Confederacy's economy and labor force, weakening their ability to sustain the war. Additionally, by allowing African Americans to enlist in the Union army, it bolstered the Union forces with more manpower, which was crucial in the later stages of the war. Overall, the strategic impact of emancipation played a significant role in the Union's military success during the Civil War.

Emancipation made military sense as whoever would have been the first to free African-American Slaves could have relied on their support in the war, which in turn have handed thousends of soldiers to that side.

On balance, I believe so. As a recruitment mechanism it made perfect sense to have former/ fugitive slaves leave the Confederate states and volunteer for a cause in which they had a personal stake. The US gained independence by pitching smallholders fearing for their lands and families against British-employed mercenaries. Nineteenth Century warfare was predominantly a numbers game. 186 000 extra soldiers was a powerful instant fillip at a point when initial passion was cooling among the volunteers. Many of the generals on both sides looked to the recent shambolic performance of the Russians in the Crimea but also at Russia's initial advantage with naval strategy and how Anglo-French boots-on-the-ground led them to squander this. That so many Black recruits went into the navy (estimated at 19000) indicates that this lesson had been learned by the Union. However, infantry won land wars; naval power helped prevent losing them.

The key advantage the Emancipation Proclamation gave to the Union was simply that the largest numbers of Men of Colour (as the recruitment posters called them) to sign up immediately were in frontline states - South Carolina and Tennessee in the first few months. It could be argued, if specific examples or numbers were provided, that the ill-treatment by Black Union POWs (known to be worse than for Whites) was a motivational force among the nascent USCT to fight harder.

What made less sense, from as objective a view as 21st Century hindsight allows, is the reluctance of some officers to pay, equip or deploy these regiments or individuals equally. Black soldiers got less daily pay and had to buy their own uniforms. In concert with the Emancipation Proclamation came the first substantial move to pay the Union forces (hitherto volunteers, as much motivated by the need to save the Union as any abstract or personal interest in slavery per se); the Proclamation came on New Year's Day, the professionalising of the army began in March. Each of these was a morale boost at a point when the war was starting to seem interminable and both sides were making grotesque errors.

A disproportionate number of African Americans (and this term definitely applies to the Union troops, just ask Frederick Douglass) were in ancillary posts (cooks, messengers and so on) but this may have offset the dwindling recruitment among White men - the glory-seekers forced into those roles were, anecdotally, the most likely to desert. (The available statistics here are, from memory, ambiguous but this seems to be a consensus among memoirs, fiction and subsequent accounts.) Although some tried to argue that the relatively high rates of ill-health and illiteracy rendered the Black recruits less useful. A more pernicious tendency seems to have been the reluctance of Union soldiers who were notionally against slavery to serve alongside PoC troopers; had the armies promoted troops solely on merit, some might have been made to serve under a Black officer, but we're now into the realm of counterfactuals and you wanted a relatively quick answer.

Forgive the brevity of this answer and lack of detailed citations: I'm living out of suitcases at the moment. You may want to start here: https://www.archives.gov/education/lessons/blacks-civil-war https://department.va.gov/history/100-objects/object-54-recruiting-broadside/ https://www.loc.gov/classroom-materials/united-states-history-primary-source-timeline/civil-war-and-reconstruction-1861-1877/african-american-soldiers-during-the-civil-war/ and consider this as back-up https://americanexperience.si.edu/wp-content/uploads/2014/09/Emancipation-and-the-Civil-War_.pdf Cornish, Dudley T. The Sable Arm: Black Troops in the Union Army, 1861-1865. New York: Longmans, 1956. YOu may want to try this: https://lsupress.org/9780807174326/emancipation-the-union-army-and-the-reelection-of-abraham-lincoln/ which I don;t think is entirely reliable but is a handy corrective to the way hindsight makes Union victory and full Abolition seem inevitable and inextricable.