slave (n.) c. 1300, sclave, esclave, "person who is the chattel or property of another," from Old French esclave (13c.) and directly from Medieval Latin Sclavus "slave" (source also of Italian schiavo, French esclave, Spanish esclavo), originally "Slav" (see Slav); so used in this secondary sense because of the many Slavs sold into slavery by conquering peoples.
Indeed, which is interesting, because “Slav” comes from the word for “speech” or “word”. In Polish, for example, “word” is “słowo”, whereas “Slavs” is “słowianie”. Germans are “niemcy” from “niemy”, meaning “mute”. So to be a Slav meant to be intelligible and able to speak, whereas Germans were unintelligible or “mute”. Compare this to “barbarian”, from the Greek, which comes from the sound of the unintelligible speech of foreigners to the Greek ear (“barbarbar”, almost like “blah blah blah”).
The first part makes sense, the second one doesn't: where would that 'c' come from? No such thing as Sclavic people, never was.
Nevertheless, I'm not sure if there is any reflection on the fact that mediterranean pirates (and nomadic tribes further east) plundered Balkans for captives and southwestern europeans then bought them. And that, perhaps, was a suboptimal way to behave.
As for why there's a "k" there - Greeks would naturally adapt any foreign word to the phonotactics of their language, which does not have "σλ" as a valid syllable onset, but does have "σκλ".
Slav (n.) "one of the people who inhabit most of Eastern Europe," late 14c., Sclave, from Medieval Latin Sclavus (c. 800), from Byzantine Greek Sklabos (c. 580), from a shortening of Proto-Slavic sloveninu "a Slav"* -- https://www.etymonline.com/word/Slav
One of your other comments sounds like you agree with the etymology jefftk cited? But that says the exact same thing. The word slav was first, and the word slave was made from it.
Something being "propaganda" and "revenge" does not stop it from being the origin of a word.
Edit:
Oh, you elaborated slightly. > It's not derived, it's used - first by defeated.
That's what derived means. They took a word and used it to make a new word.
That's nonsense, not true and a libel.
That words are INDEPENDENT:
slave doesn't mean faming, people names:
Bogusław mean Godfaming - not slave of god,
Mieczysław mean Swordfaming - not slave of sword.
https://www.etymonline.com/word/slave