I've previously argued that Lincoln, more than anyone else, wanted the Union to get over the "late unpleasantries" as quickly as possible. Much like Sherman, he was eminently practical. His goal was to restore the Union by any means, and to decisively put an end to the secessionist impulse that had essentially been growing in the Southern USA since Jefferson's Virginia and Kentucky Resolutions of 1798 and 1799. But once that was achieved, he wanted to turn over a new leaf. (I compare this to Sherman, because he prosecuted the war by all available means, but as soon as the peace was concluded, he supported rebuilding the South and quickly ending any military occupation.)
If the murder plot is completely avoided, I see the radical Republicans being completely side-lined, and the federal government actively aiming for reconciliation. This means that the South gets off without altogether too much Reconstruction, with the understanding that they don't make any more fuss. In this way, the whole secession will become something of an embarassment that the South will want to forget. The humiliation of Reconstruction led to the glorification of the "Lost Cause". Avoiding that means that everybody just wants to forget that it ever happened at all.
The dark side of that is that, even though slavery gets abolished, the Southern states are otherwise free to still do whatever they want. Sure the OTL Amendments are still going to happen, but the South will be putting voting qualifications in place before you can say "racism!" and they'll be 100% Democrat-governed again within the year.
On the other other hand, the absence of resentment over being humilated and threatened by "the black Republicans" means that you presumably won't actually see Jim Crow. Instead, there will be more of a low-level oppression and side-lining of non-whites. The South will just segue into that right after slavery is abolished, and I don't expect anyone to do much of anything about it.
The USA, in a federal sense, is politically more stable in the following period. Things just "go back to normal". The Democrats are more of a factor on the federal level again much more rapidly. The CSA leaders, mostly, will actually be considered embarrassing figures. They won't ride into prominence on the back of anti-Reconstruction discontent. A new generation of Democrats will just pretend none of it happened, and the Republicans will let them, because they want it all behind them, too.
Personally, I think this would ultimately be an improvement compared to OTL. It removes radicalism and normalises political relations. This means that injustices that exist in this ATL can in the end be resolved with less fuss. (But then, I am known for my stance that revanchism is a mental illness, and that reconciliation is ultimately the healthier approach -- even if it involves a lot of clenched teeth. You can unclench your teeth later! You can't bring the dead back to life if you let it come to conflict!)
