To be blunt, all that the freedman would have gotten from Greeley and the Liberal Republican's platform in 1872 was the end of Reconstruction and Jim Crow come early.
Before explaining the thought-processes behind abolitionists and African Americans in deciding to overwhelmingly vote Grant over Greeley, I'll first provide some context for why Greeley and the Liberal Republicans challenged Grant. At first glance, the Liberal Republican party appeared to have been a who's-who of prominent abolitionists, containing many highly respected anti-slavery advocates such as Charles Sumner, Salmon Chase, Cassius Clay, and of course Greeley. The party formed from a coalescing of grievances against Grant's administration, including its corruption, blatant cronyism in the civil service, and imperialism. Yet perhaps the most prominent issue between the two Republican parties was over Reconstruction. Despite the Liberal republicans including many former Radical Republicans, the party espoused a classical liberal ideology of small government and felt that, with the passage of 13-15th amendments, Reconstruction had completed its goals of enfranchising and empowering African Americans while burying the Confederacy. Greeley and others believed that the continuation of Reconstruction only sought to further radicalize white southerners, and the Liberal Republicans sought to reconcile the North and South, advocating for amnesty (enfranchisement) of former Confederates and returning local governance back to the Southern states. It must be noted that, besides the Liberal Republican's contempt for interventionist government and the suspensions of rights that occurred during Reconstruction, Greeley and others did not believe in the notion that African Americans were suited to running the post-war governments of the South. The Democratic Party, essentially disempowered in the South, understood that the Liberal Republicans were the only viable opposition to Grant, and nominated the Liberal Republican ticket at their own convention.
This does not mean that Greeley and the Liberal Republicans did not attempt to court the abolition and freedmen's vote. The Liberal Republicans frequently referred to their abolitionist credentials and framed Greeley as one of the oldest friends of the African American population. However, most other prominent abolitionists, and many of the most influential African Americans of their time, distrusted Greeley and lent their support to Grant. William Lloyd Garrison attacked the Liberal Republicans for taking credit for being early adopters of abolitionism and assailed Greeley as a compromiser on slavery who supported the emigration of freed slaves to Africa. Fredrick Douglass himself penned a letter to his fellow "Colored People" deriding the keynote speaker at Greeley's nomination for stating that "one of the objects of the movement the “overthrow of negro supremacy." John Mercer Langston, a Black politician and law dean of Howard University, warned that Greeley's victory would be a return to State's Rights and the end of enforcing civil rights. All of these men and many others opposed the Liberal Republican alliance with Southern Democrats, and all pointed out Grant's exceptional track record in aiding the freedmen of the South, from vigorously combatting the Ku Klux Klan to allowing for "a negro in the U.S. Senate in the seat of Jeff[erson] Davis."
These apocalyptic claims by Grant's supporters were not unsubstantiated. The succession of Radical Republican governor Brownlow in Tennesse by his moderate Republican lt. gov Dewitt Clinton Senter allowed for the re-enfranchisement of former confederates and the establishment of the poll tax in the state. And in 1868, Democratic state legislators in Georgia allied with white moderate Republicans to expel all 33 non-white state legislators from office, with many having to flee the state to avoid lynching. When one compared the militant Reconstruction policies of Grant with the compromising and conciliatory platform of Greeley, it was obvious that the freedmen would stay with the Republican Party that had fought so hard to grant them their rights in the first place.