I can't recall hearing this saltiness quote before. But I don't read the NT that often. And I sure can understand why preachers would stay away from this quote , so it is not as popular as other quotes.

If NaCl (solid) is somehow mixed with CaCO3 (solid), then exposing this mixture with water will more readily dissolve the NaCl than the CaCO3. The water leaves with NaCl. Then this "salt" has lost its flavor.

This could be somewhat reliable as salt mines are often next to limestone. The mining processes of antiquity probably mined a bit of both. But that is only a quick hypothesis that deserves further research.