If you’ll read the articles of secession, they make it plain why the offer to let slavery continue in the south was a non-starter for them. They state clearly, that the thing they could not accept was Lincoln’s refusal to let slavery extend into the western states. They said that his refusal to continue the previous practice of bringing in states two-by-two, one slave and one free, would over time upset the balance of power and the slave system would be thereby doomed eventually.
You are right, that the common southern soldier did not fight for slavery, he fought for his state. The leadership, however, is quite clear why they demanded independence. In their own words, the slave system, which they considered an entirely moral system, and more importantly, control of the western territories.
They were already outvoted in the Congress. This resulted in the vast bulk of their economic output flowing through the hands of New York and Washington DC, while they themselves were left with whatever remained. So long as the Northern coalition controlled congress, they could do nothing about the situation. When it was no longer possible to get newly created allied states to join the Southern voting bloc, they realized they would be forever relegated to being the milk cow for Northern interests.
They decided they wanted out. The deal wasn't working for them.
They were paying almost all the taxes funding Washington DC.
Here is an article of secession that I urge you to read.