I don't understand this point. Universal suffrage would in theory have helped Republican prospects in the South, if freed slaves voted Republican. But would they have constituted a majority in any Southern State or congressional district? In the two Southern States where Lincoln was on the ballot in 1860, he got only 1% of the vote. Even in Maryland he only got 2.5%.
Moreover how would eliminating the 3/5th clause help Republicans in the South, other than making them even more popular with freed slaves? The mechanical effect of that change would just have increased Souther representation in Congress, rather than helping either Republicans or Democrats in the South. Unless the author is projecting the impact of redistricting?
The 14th amendment tried to force the Southern states to grant voting rights by penalizing them if they did not let black men vote--if there was a large number of black people, the state could lose some of its seats. The Northern states had such small numbers of black people that even if they didn't let black men vote (which was still true in 1866 in some cases) they wouldn't lose any seats (according to Gary Gallagher in 1860 the non-slaveholding states were 98.8% white).
Once slavery was abolished, the three-fifths rule became moot. It would still be effective, but would apply to a nonexistent group of people. You could just as easily say that space aliens only count as three-fifths of a person.
Yes, the state of South Carolina was majority black while Alabama and Mississippi were around 50% black. In addition, many congressional districts through the cotton belt were majority black then just as they are now.
In several southern states, blacks were a majority or close enough to win free elections. In a less polarized South, Republicans could have presumably drawn enough white votes to win in several more states.
The point, I guess, is that the South would have had more representatives in the House after the war than before, due to the 4M newly counted as 5/5 rather 3/5. But I think he’s stretching a point to assume the additional congressmen would be GOP.