Cannot agree with you that Lincoln was a terribly good politician. He was a VERY good human being, a brilliant president, and a terrific war leader. His political skills were somewhat limited (take a long look at his manuvering during the runup to the 1862 elections for instance), but he made the most with what he had.
The real problem was that post ACW just wasn't a 'winnable' scenario for any president, no matter what his skills. The Radical Republicans (RR) had more than enough votes to push through just about anything they really wanted, and what they wanted (for many reasons, ranging from real conviction to simple greed and lust for power, all the way to plain vindictiveness) was the utterly demolition of the South as a political, economic, and cultural component of the US. The idea of 'state suicide' (i.e. the seceding states had essentially 'comitted suicide' by leaving the Union, and thus had no legal existence upon their reconquest, the brainchild of Hamblin), is a perfect example of this sort of thinking. No president who wasn't willing to go along with this policy was going to avoid a truly awful beating at the hands of a bunch of unscrupulous and vindictive ideologues.