To suggest that the States were in rebellion against the Union is to concede the southern claim that the Civil War was a "War Between the States." People still in rebellion--sadly, some of them posting in this forum--continue to make this claim.
That the State governments and people of those states were in rebellion against the Union makes it difficult for me not to concede their point. Was the American Revolution a rebellion against England or was it simply the people living in some English colonies that were rebelling against the English crown? What's the difference?
But I don't think that conceding that point makes it impossible to agree with Lincoln's response. Yeah, they were in rebellion. Yeah, the Union decided not to let them go without a fight for a variety of reason. It's not as if we didn't take or keep other parts of the country by force both before and after the Civil War.
I think I agree with your general statement.
The South was in rebellion, including the governmental apparatus of all of the Southern States. The CSA was a democratic republic that had real juridical existence, and although not recognized as such by the USA de jure, it was certainly recognized de facto by the hard fight against it, as well as the honors of war paid to captured officers (in some cases), etc.
The South had no RIGHT to be in rebellion, or to become a separate government, according to the United States, and so the USA conquered the CSA.
The issue wasn't resolved by logic, or argument, but by war, and for that very reason the lines of legal argumentation break down, and you have anomalies like official acts undertaken by states that had not yet been readmitted to the Union.
You also had other anomalies, like an income tax without apportionment during the Civil War years. One can't find a nice, neat legal conclusion, because the war cuts through it all like a jagged edge. War itself was against the rules, so the fact of it, which is the central fact of the whole era, means that any search for a nice, round and complete legal explanation for either the war itself or the aftermath fails.
The 13th Amendment was adopted by states that had not yet been readmitted to the Union yet. It was the law and the Constitution anyway, because that was ultimately what the war distilled down to, and the victors imposed the rule which stuck.