Short answer is that if you're using the Experience Cloud ID service, the cookie should be all you need. if the visitor ID cookie gets cleared, it's treated as a brand new visitor, which means that any historical eVar persistence doesn't matter. The 'never expire' eVar is tied to the old visitor ID and has nothing to do with the new one.
Additionally, if cookies are cleared, chances are close to guaranteed that cache is cleared too, wiping any visitor data in local storage. In my opinion it's overkill, and that the minimal value gained by copying visitor ID info to local storage doesn't outweigh the complexity and time implementing it. You'll still ultimately get the same trends in reporting.