Right, the salt will actually cause the freezing point to be depressed--hence why salt is added to icy roads--and the slurry will actually be colder than 32 degrees if salt is added.
Just adding ice to water is insufficient guarantee that equilibrium exists, obviously if left out in a room, the ice will all melt and water may never get close to 32, hence the suggestion that the melted ice occur near freezing as in a fridge.

Good point about the additional point at mash temps where accuracy is important. For not too much money, narrow range thermometers can be bought covering the zone of interest with better resolution. Good digital thermometers these days are not bad at all, I had one specced againt a standard and it was +/- 0.6 F from 110 to 170. No worry about breakage!