Pro-Democratic newspapers in the North did attempt to ‘draft Lee’ for the 1868 election.
Here would be a sample from the New York Herald July 1st 1868.
"But if the Democratic Committee must nominate a soldier — if it must have a name identified with the glories of the war — we will recommend a candidate for its favors. Let it nominate General R. E. Lee.
Let it boldly take over the best of all its soldiers, making no palaver or apology. He is a better soldier than any of those they have thought upon and a greater man. He is one in whom the only genius of this nation finds its fullest development. Here the inequality will be in favor of the Democrats for this soldier, with a handful of men whom he had moulded into an army, baffled our greater Northern armies for four years; and when opposed by Grant was only worn down by that solid strategy of stupidity that accomplishes its strategy by mere weight. With one quarter the men Grant had this soldier fought magnificently across the territory of his native State, and fought his army to a stump.
There never was such an army or such a campaign, or such a General for illustrating the military genius and possibilities of our people; and this General is the best of all for a Democratic candidate. It is certain that with half as many men as Grant he would have beaten him from the field in Virginia, and he affords the best promise of any soldier for beating him again.
Lee declined and even threatened to resign Washington College due to the verbal attacks that were being carried out by members of board of Governors on Grant. Two months after Grant’s inauguration he was invited to the White House for talks and on that topic it’s gets interesting as there are four or five versions of what was said.
The best you were going to get post war unless another war happened such as with France or Spain was a notable ex-Confederate Vice President who gets a promotion if/when the head of the WH bites it.
The issue with anyone going too deep on one side as with some Confederates who cozied up to the Republican Party as they were actually hated worse then the northern Republicans among Democrats. You needed someone able to play the game of being political, but apolitical.
Having war clouds loom in 1867-1868 with Britian or France could have gotten a national unity ticket of some kind.