Why stop at adding fluoride? I think we should be adding small amounts of lithium to public water. The public mental health benefits could be enormous.
The US does already provide iodine supplementation, just via table salt and not water (I suspect because it's a more convenient way to deliver such supplementation)
Some of us who eat Japanese food (which is often very high in iodine, mostly from seaweeds like kombu) appreciate that we aren't forced to overdose even further!
If you eat "too much" salt you get thirsty, drink more water, piss out the salt and restore balance. The myth of salt being unhealthy was funded by snack food companies who knew that if they scared people away from adding salt to their normal meals they would still crave salt and likely turn to snack foods to satisfy that craving. Adding salt to your home cooked meals and abstaining from junk food is the healthy choice.
Salt is essential to neurological well-being. As you say, as long as your kidneys are healthy, you can have much more salt in your diet than is currently recommended. (Obviously not too much at once, as that can kill you.)
Though not Kosher salt, or most "premium" salts, or any of that Himalayan salt. Be careful that on your quest for the most gourmet salt you don't prevent your children from getting enough iodine!
Throw in some iodine too for good measure.