Not really. It evidences its presence as a major issue, but not "the" issue. If you look through the legislation of that session, you will find vehement southern opposition to several other matters both slavery related and not. One of the most contentious pieces, for example, was the Morrill tariff bill. Things were also still brewing from the debate over the Homestead bill from the previous summer. Senator Robert MT Hunter, for example, summarized his grievances with the Morrill tarriff bill in one of the longest speeches of the session. It concluded with the statement "I know that here we are too weak to resist or to defend ourselves; those who sympathize with our wrongs are too weak to help us; those who are strong enough to help us do not sympathize with our wrongs, or whatever we may suffer under it. No, sir; this bill will pass. And let it pass into the statute-book; let it pass into history, that we may know how it is that the South has been dealt with when New England and Pennsylvania held the power to deal with her interests."
Such views were characteristic of the southern grievances on practically every issue they opposed - it was not all "you're going to abolish slavery." It was rather that "you have complete control of the government and are using that control to push policies that put our region and economy at the mercy of northern interests."
It was the only major legislation passed to try to head off secession and it directly addressed the question of slavery to try to assure the South there would be no abrupt change of the status quo.
The reason it was the only legislation passed to head off secession was due to northern obstructionism in a small "radical republican" crowd that used parliamentary devices to tie up practically everything except for the bills they wanted. For all practical purposes, that amendment was the only measure they could not muster the strength to halt (though they did indeed try, and that is why it barely passed with only the slimmest of margins). If you want a glimpse of just how obnoxious and obstructionist the Charles Sumners of that session were, you don't even need to read the southern complaints against them. The statements and writings of the leveler headed members of the GOP, and even some of its more outspoken members, express equal disgust and frustration with Sumner and his following. Thomas Corwin, Charles Francis Adams, and even William Seward put much of the blame for the inability of congress to reach a compromise on Sumner. If anything, he is one of the major reasons why many states seceded before Lincoln's inauguration and why legislative attempts to keep them in the union failed.