I think you're on the right track that L/T/Y dwarves (brown dwarves) should have cool enough atmospheres to have NaCl in them. I don't know what references to go to say for sure, though.

One of the problems isn't just that the salt molecules need to be warm to emit (that's true), but that the wavelengths at which we see their radiation are tough to observe in stars and planets. The detection we reported was in a disk - which is very, very big compared to a star or planet, and so we could see it at radio/millimeter wavelengths. We generally can only detect stars themselves at optical and infrared wavelengths, and it turns out that NaCl and KCl don't have many transitions at wavelengths we usually observe (e.g., https://ui.adsabs.harvard.edu/abs/2014MNRAS.442.1821B/abstract). Most of their strong emission/absorption lines are at >=26 microns, which is just at the edge of what JWST is capable of observing with its MIRI instrument. No other telescope has observed at these wavelengths with enough sensitivity to pick up salt molecules. I think there's some possibility JWST will detect salts in either hot jupiters or brown dwarves, though; there are weaker salt lines covering JWST's whole range. The trick is, there are lots of other molecules that could obscure the salts in an atmosphere - I'm not sure whether we'll be able to identify the molecules cleanly. It's a much easier job at radio wavelengths.