Does water "go bad" in this sense?
It doesn't go bad, but it does change the taste.
When water is just sitting there, water evaporates, but most things dissolved in it don't. Then, each time you boil it, the steam causes additional water to escape leaving the same amount of dissolved stuff in there. So, the concentration of dissolved stuff keeps going up.
Dissolved oxygen also decreases when you heat or boil water, changing the taste.
Finally, even if your kettle is stainless steel, it can eventually rust.