From what I've heard, save files in-browser are a mess of jumbled data that can really only be interpreted by the browser itself, so it's not really stored anywhere a person could access; if you want an easily-accessible save file, the "save to file" option is really your only option.
Unless you're saying that even the game itself can't find the saves. That can happen if you're running two or more instances of the game at once, but if not, then yeah, wiping various browser data (which the browser can do by itself if you don't turn those settings off) is the most likely culprit. Again, save to file to avoid that in the future.