I think the “illah” in Allah is cognate with the Hebrew El or Elohim, found in the Bible as a word meaning “God.” Arabic and Hebrew are related languages (seen also in “salaam” and “shalom” both meaning “peace”).
Arabic “illah” definitely seems cognate with Hebraic El.
I’m sure you know it, but will just repeat here for others - you are correct that “salaam” and “shalom” both mean peace, but “Islam” is not from “salaam” but means “submit”