Here is why I think it is possible for Grant to have won in 1880 (though not 1876, when as I said, I think no Republican but Hayes could have carried Ohio and therefore the Electoral College):
We have to remember that not only were Gilded Age elections close but that party allegiances were firmly held. There were very few "floating" or "swing" voters; "weak" candidates for a party did almost as well as "strong" ones. It is therefore reasonable to assume that Grant would have won all the states Garfield did except the really close ones.
Now if you look at
http://psephos.adam-carr.net/countries/u/usa/pres/1880.txt you will see that there were not that many narrowly-for-Garfield states in 1880. The only states Garfield won by less than four percentage points were Connecticut (6 electoral votes), Indiana (15), New York (35), and Oregon (3). In addition, he won one electoral vote from California. A Republican candidate could have lost all those electoral votes except New York's and still have won 189-180.
So the real question is whether Grant could have carried New York. My answer is that it was possible though by no means certain. Garfield's 21,000 vote margin there looks narrow but compared to Cleveland's 1,000 vote margin in 1884
http://psephos.adam-carr.net/countries/u/usa/pres/1884.txt or even Harrison's slightly-over-14,000 vote margin in 1888
http://psephos.adam-carr.net/countries/u/usa/pres/1888.txt it is not that slight.
One reason Garfield carried the state: Tammany's "Honest John" Kelly insisted on running an Irish Catholic, William Grace, for mayor. Grace narrowly won, but cost the Democrats many Protestant votes, and may have cost Hancock the election. Still, what was important for Kelly was to have a mayor who would let Kelly keep control of city jobs. Having a Democrat in the White House would have been a dubious advantage for Kelly because at best a President Hancock would divide New York's federal patronage between Kelly and Kelly's rival Tilden; and at worst, Hancock might have aligned himself with Tilden or built a rival anti-Kelly power base in New York. It may be that Tammany Hall just didn't want a Democratic president at this time. And this fact would benefit any Republican nominee--including Grant.
Another thing that would benefit Grant as it did Garfield: Hancock was not a great candidate. He gave an interview where he called the tariff a "local issue" which gave Republicans a chance to pounce on his supposed ignorance, as in Thomas Nast's famous cartoon showing Hancock asking "Who is Tariff, and why is he for revenue only?"
Now it is true that there would probably be some reform-minded Republicans who would simply refuse to vote for Grant. These were the so-called Independents who had been Liberal Republicans in 1872 and would become pro-Cleveland "Mugwumps" in 1884. Yet Cleveland only won New York by 1,000 votes that year, despite the defection of the Mugwumps, despite the fact that some pro-Grant, pro-Conkling "Stalwarts" refused to vote for Blaine, despite the "rum, Romanism and rebellion" controversy , despite bad weather on Election Day in upstate New York, despite the fact that Blaine had ethical problems of his own, and despite the fact that Cleveland had a home-state advantage in New York (remember that the only Democratic presidential candidates to carry New York between Pierce and Wilson were all from the state--which Hancock was not). So I would by no means rule Grant out in 1880. If to the reformers he meant the scandals, to many ordinary voters he still meant Appomattox...
