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.
They had plenty of opportunity to do so, and never took it.
In 62 or 63 if the CSA had agreed to emancipation, even delayed and compensated, the Brits would have fallen all over themselves recognizing and probably allying with them. The RN would have broken the blockade and the South wins the war.
This exact scenario was proposed by Patrick Cleburne, one of the very best southern generals, and caused a furor which was repressed by Davis himself. It also kept Cleburne from being promoted for the rest of his (cut short by being shot) career, despite the army's desperate need to competent senior leaders.
Basically, the South had a choice between losing the war in a desperate attempt to keep slavery going, or scheduling future emancipation and winning their independence.
They took option A. (They chose poorly.)
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Maybe that's going too far, but actual slavery would have been replaced by forms of labor-force control that weren't radically different from slavery, as was indeed the case in the history we know.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.
Here is that letter from Jefferson Davis to Pope Pius IX.
Note that it says nothing remotely resembling what you report.
Here is a more detailed report on letters among the Pope, Davis & others, and their importance.