I kind of like thinking about those two words set against each other, though!
Sounds like “adagio” is a contraction of “ad agio” as in “at leisure” in Italian, while the usual references suggest that “adage” might have a couple possible derivations. The main one being that it comes through French from the Latin “ad” + a form of “aio,” or “I say,” so in the straightforward sense “a saying”; another etymlogy positing a root in “adigo” as in “drive, force.” [0]
Sounds like “adagio” is a contraction of “ad agio” as in “at leisure” in Italian, while the usual references suggest that “adage” might have a couple possible derivations. The main one being that it comes through French from the Latin “ad” + a form of “aio,” or “I say,” so in the straightforward sense “a saying”; another etymlogy positing a root in “adigo” as in “drive, force.” [0]
[0] https://www.etymonline.com/word/adage