This is the sole constitutional remedy for the sort of abuses Jim Crow States were imposing on the black population in the 40's and 50's. There were multiple remedies, of course: one, black citizens emigrated north, signed on with city machines in big States like Illinois (home of Emmett Till) and New York, and exerted influence at the federal level back on their home States.
Two, the XIV Amendment remedy of reduced representation, of which the House of Representatives would apparently be the arbiter.
Three, another constitutional amendment to provide further remedies along the lines of the 1965 Voting Rights Act.
Four, litigation under their state constitutions against the abuses complained of, up to the level of the U.S. Supreme Court to see whether SCOTUS would bind voting rights to the States as it had done other federal rights, as had been done in Brown vs. Board of Education and in the case of the poll tax, which was found unconstitutional.
Notice that I said that the VRA would require amending the Constitution, to be constitutional itself.
As you are no doubt aware, no attempt was ever made to implement this provision of the 14th, so we don’t know how it would have administered. The amendment provides no mechanism for adjudication or administration of the provision.
Presumably, as you say, it would have been by the House, which is the only institution directly affected anyway.
A very awkward provision, arising out of an attempt to square the circle of respect for state determination of who had the right to vote with federal discouragement of disenfranshisement of black citizens.