Bradford opens his article with the following:
"The growth of a [Robert E.] Lee legend is greatly to be deplored, most of all by Lee's warmest admirers. ‘ One may search in vain for any defect in him,’ says one of the latest historians of the war. ‘Indeed, the perfection of Lee becomes somewhat oppressive. One would welcome the discovery of a shortcoming in him, as redeeming him to humanity.’
This is unfair, but not unnatural, when one considers the attitude of Lee’s Southern admirers.
‘He was never behind time at his studies, never failed in a single recitation, was perfectly observant of the rules and regulations of the institution,’ says an old teacher. ‘Throughout his whole student life he performed no act which his pious mother could not have fully approved,’ says another. I do not believe this is true. I hope it is not true. If it is true, it ought to be concealed, not boasted of.
This is the sort of thing that made [George] Washington odious to the young and remote from the mature for generations. ‘In all essential characteristics Lee resembled Washington,’ says Mr. Rhodes, with much justice.
But we know that, in spite of ill-judged idolatry, Washington was not a prig. Neither was Lee, but a man, of warm flesh and blood, like the rest of us. No one could have had his large and tender sympathy for human weakness who had not known human weakness himself. Above all, those who knew him, from the common soldier to the president of the Confederacy, bear universal testimony that Lee had charm.
Now, no prig ever yet had charm. Therefore I refuse to believe that he said — at any rate, in those words — to Magruder in Mexico, ‘I am but doing my duty, and with me, in small matters as well as in large ones, duty must come before pleasure.’"
We also know that Union Gen. Ulysses S. Grant was also elected to the U.S. Presidency in 1869, succeeding Andrew Johnson, who had succeeded Abraham Lincoln as President in 1865. With Lee being popular among Southerners, was there anyone who ever suggested he run for President?
Subsequently, were there any descendants of Lee who were encouraged to run for political office?
So I saw this question yesterday, and I was decently confident the answer was "No" already, but have spent a bit of time yesterday and today going through some of the books I have handy before being able to say that with enough confidence to write it here. In plainest terms, there is nothing I can find which suggests any sort of attempt to draft Lee for a Presidential candidacy in any meaningful way. The most involvement I am aware of, at all, in Presidential politics, was his endorsement of the 1868 Democratic candidate Horatio Seymour, where his signed a public letter, alongside several other former Confederate figures, at the behest of William Rosecrans, a former general in the Federal forces, and a Democratic backing Seymour. Along with appending his signature to the letter, Lee was active in recruiting other fellow traitors in doing so as well.
And I use traitor there purposefully as it is important to bring up two key caveats here which make it so unlikely that Lee would have been considered for a Presidential run. In the first, he was a traitor, morally speaking. It was well understood that for any chance at an electable candidate, Democrats needed to run Northerners for the foreseeable future. Seymour was a New Yorker. In 1872, they didn't even run a candidate and instead the Democrats backed the Liberal Republican candidate, Horace Greeley, in the Republican political split. After that we see New York, Pennsylvania, New York, etc. Wilson was the first Southerner (and even then, while from Virginia, he had long lived in New Jersey) to make it to the White House, and it wasn't for a century, with Texan LBJ, that a Democratic born and living in a former Confederate state was President! The point being that it was generations before Southern Democrats had potential to be national candidates.
Even that aside though, Lee himself couldn't be the candidate. Again, he was a traitor, this time legally speaking. While there had been amnesty for most soldiers of the Confederacy, Lee's citizenship was never restored in his lifetime, due to political factors, and it wouldn't be for over a century that it happened, of course on a purely ceremonial level. But in any case, due to this fact, Lee was ineligible to hold public office even if he had wanted to, and if there had been support for it. How much that factored into the lack of even proposing it, is hard to say, but certainly it was a clear hinderance. Perhaps we can hypothesize a future where Lee gets it restored, and contests in 1876 or something, instead of being dead by then, but that is a counterfactual far beyond what is reasonable to trace out here.
As for your final question, to be sure, the Lee name held great sway in the years after, but not for Presidential ambitions. His nephew, Fitzhugh Lee - himself a war hero in the eyes of Virginians - was able to gain the governorship of the state, and Robert's son William Lee (whose middle name was also Fitzhugh...) was elected to the House of Representatives, where he served until his untimely death. So certainly, the popularity of Lee was a boon for familymembers with political ambition, and his insurrection against his country was no hinderance to it either in the eyes of Virginians, but for a confluence of reasons, political office, let alone the Presidency, was never in the cards for Lee.
Further reading
Alan Nolan's Lee Considered
Gary Gallagher's Lee and his Generals
Link to Gamaliel Bradford Jr.'s 1910 article: https://www.theatlantic.com/magazine/archive/1910/12/a-heros-conscience-a-study-of-robert-e-lee/644249/