Slavery was "headed for extinction" in large part because the Confederacy lost the war. Had it won, things might have looked different.
Brazil wouldn't have felt isolated as a slave state and slavery might have lasted longer. The same would be true of the remaining Spanish colonies, Cuba and Puerto Rico. Slavery might even have been restored in some countries or territories.
Of course, nobody imagines that slavery might have endured down to the 21st century, but any "time table" of emancipation would be affected by the result of the war. Slavery could have endured until mechanization made hand-cotton picking unprofitable in the 20th century.
Maybe that's going too far, but actual slavery would have been replaced by forms of labor-force control that weren't radically different from slavery, as was indeed the case in the history we know.
I read somewhere that there was a letter in the Vatican archives from Jefferson Davis where he acknowledged that slavery was on its way out and that he would certainly trade it for Southern independence.
Of course, he would say something like that, since he was looking for foreign support. If he did in fact say it, it doesn't mean he was sincere.
As you know, many of the (delusional) southern extremists planned to expand their slave empire to the South, taking the Caribbean and (depending on the degree of delusion) everything to the Isthmus or Cape Horn. I’m unclear whether they planned to enslave the existing inhabitants or start the slave trade back up in a big way.
They were utterly delusional because the only way, at the time, to invade these areas was by sea, as the US found out during the Mexican War.
The RN, and for that matter the US Navy, would never have permitted such an expansion of the slave empire or a restart of the slave trade. And having no navy, the CSA would have been unable to do anything about it.