Again, slaveholders of the South regarded the Corwin Amendment as inadequate because it did not remedy the fugitive slave problem or the risk of slave revolts. Secession arguments and documents often expressly renounced the US constitution.
Again, Lincoln offered strengthened fugitive slave laws. Secession would not fix the fugitive slave problem, it would make it worse. The US would be a different country and thus under no obligation to return escaped slaves.
"But secession, Lincoln argued, would actually make it harder for the South to preserve slavery. If the Southern states tried to leave the Union, they would lose all their constitutional guarantees, and northerners would no longer be obliged to return fugitive slaves to disloyal owners. In other words, the South was safer inside the Union than without, and to prove his point Lincoln confirmed his willingness to support a recently proposed thirteenth amendment to the Constitution, which would specifically prohibit the federal government from interfering with slavery in states where it already existed." (Klingaman, Abraham Lincoln and the Road to Emancipation, pp. 32-33)
Nobody could refute what he said about how slavery was safer in the union than outside it. Lincoln wasn't even the first man who pointed this out.
In the debate in Congress on the resolution to censure John Quincy Adams, for presenting a petition for the dissolution of the Union, Mr. Underwood, of Kentucky, said: "They (the South) were the weaker portion, were in the minority. The North could do what they pleased with them; they could adopt their own measures. All he asked was, that they would let the South know what those measures were. One thing he knew well; that State, which he in part represented, had perhaps a deeper interest in this subject than any other, except Maryland and a small portion of Virginia. And why? Because he knew that to dissolve the Union, and separate the different States composing the confederacy, making the Ohio River and the Mason and Dixon's line the boundary line, he knew as soon as that was done, Slavery was done in Kentucky, Maryland and a large portion of Virginia, and it would extend to all the States South of this line. The dissolution of the Union was the dissolution of Slavery. It has been the common practice for Southern men to get up on this floor, and say, 'Touch this subject, and we will dissolve this Union as a remedy.' Their remedy was the destruction of the thing which they wished to save, and any sensible man could see it. If the Union was dissolved into two parts, the slave would cross the line, and then turn round and curse the master from the other shore." In attempting to secede from the Union, the South had to be aware that they were, effectively, giving up their slaves.
Which is more rational and believable....that the White Southern population was motivated by a concern for the preservation of slavery which was not threatened on behalf of the 5.63% of the White Southern population which owned slaves, and therefore undertook to secede which would effectively mean the end of slavery OR the vast majority felt they were being economically exploited and abused by a (to them) foreign Northern majority which was constantly pushing them around and enacting taxes and economic policies which hurt the South for their own benefit?
"When the only tool you've got is a hammer, all your problems start to look like nails."
You keep trying to shoehorn "slavery" into every decision made in that era because that suits what you want to believe.
The fact the Southerners didn't take the Corwin deal is because it wasn't the main sticking point, and what they already had was way better. The fact the Northerners offered the Corwin amendment deal is because what they really wanted was the money.
Secessionists didn’t trust the Republicans. They thought Lincoln would use his appointment power to build up the Republican Party in the Border States. Eventually, slavery would be abolished in those states and the Republicans would organize in the Upper South and do the same there. The secessionists recognized the Corwin Amendment as a last minute attempt to hold the country together. The proposed Amendment did nothing to overcome their fear and their hatred of the Republicans.
The Deep South states were already gone when Lincoln took office. They’d gotten what they wanted. They assumed that slavery and their way of life would be secure as an independent nation. They weren’t going to do a U-turn and come back. And yes, the Corwin Amendment wouldn’t do anything about the fear of Northern abolitionists and slave revolts.
“Stronger” fugitive slave laws apparently means non-enforcement or repeal of the Northern states personal liberty laws. If Lincoln had given the slave states that it would have torn the North — and the Republican Party — apart. Could Lincoln have done it? Would he have? Northern states weren’t likely to repeal their laws, nor was Congress likely to deliver even stronger fugitive slave laws. Lincoln was repeating his commitment to enforce existing laws. He was also making a last minute offer to save (what was left of) the Union. The militant secessionists who controlled the Deep South easily saw through the offer. It wasn’t something Lincoln could make happen and it would have doomed his presidency if he had tried.