Cough cough... I spent a good 2 years drafting a timeline along these lines a while back, only with the trigger for successful Reconstruction being an early arrival of the boll weevil right in the middle of the Civil War, triggering massive unrest and uprisings from within the Confederacy simultaneously with the Union's advance and destroying the plantation economy enough to, combined with butterflying Lincoln's assassination, allow for successful land reform via the Freedmen's Bureau, along with general boll weevil-induced planter bankruptcy just naturally forcing the big plantations to sell off their land as their cotton crops fail.
In terms of politics I did pretty much what a lot of y'all have suggested already, that being economic empowerment of freedmen as smallholders being the catalyst for entrenching their political power within the existing economic system of the United States. The GOP, somewhat like what Nyvis proposed, ended up as essentially an overarching agrarian alliance between Midwestern farmers and Southern smallholders and freedmen under the umbrella of free trade policy and pro-rural farmer economic policy. The Populists also, as suggested, never really came into existence and as the Greenbacks died they effectively became a secondary support party to the Republicans on fusion tickets before fully merging back into them.
In terms of opposition politics I ended up having the Democrats slowly dwindle as a major party as the plantations died and GOP smallholder interests dominated the reconstructed South, gradually moving out into the West behind a wave of ex-Confederate emigres from the South onto the frontier as a secondary player. The niche of urban politics, big business, political machines in the Northeast like Tammany Hall, Catholic/European immigrant interests, and protectionism/Goldbug interests ends up filled by the Liberal Republican Party, a more long-lasting version of Horace Greeley's political vehicle from 1872 that arises from the inevitable breakaway of urban Republicans from such a rurally-focused GOP. After losing in 1872 thanks to vote splitting with the Democrats they end up coming to an agreement with the now primarily Western Democrats whereby the Democrats are effectively a Western wing of the Liberal Republicans which votes for their federal candidates on a fusion ticket in exchange for being left alone (although I was going to have them eventually merge into a party just called the Liberal Party in the 1880s).
This, along with the significant period of time in which the Republicans are supported federally in many weak spots by the Greenbacks, also has the notable effect of giving both parties an interest in avoiding OTL anti-electoral fusion laws, and fusion tickets where the major parties are supported federally by local affiliate parties become a pretty entrenched thing in American politics by the 20th century. In particular I was planning for the Utah People's Party to survive post-statehood as a Mormon interests party instead of splitting between the Democrats and Republicans thanks to the Republicans trying to court non-Mormon miners onto their rural platform even harder than OTL with continued anti-Mormonism, with the main reason the People's Party doesn't just merge into the Liberal Party by the turn of the century being that lots of people just have negative associations with the world Liberal, due to the history of the confusingly unrelated anti-Mormon Utah Liberal Party.