This is something I happen to know a little about because I researched this previously.

You are totally correct that "sal" and "salary" have an origin related to salt, and it was tied to the word "salarium"....however the following are misconceptions:

Rather a salarium was their specific "salt allowance". So they were paid using money, but the breakdown of their payment included an allowance for salt. For whatever reason, this term got generalized into "salary" (most likely it was done ironically / jokingly, like "did you get your salt yet?")

A modern day equivalence is how US people refer to dollars as "bucks". This stems from frontier settlers trading buckskins that then got generalized into "bucks" being a synonym for dollars. However, it was never common practice that people were paid in "buckskins" its just one of those linguistic memes that humans like to create.