You misunderstand,
You can create an implict auth token which’ll last for around 60 days
Or a regular user auth token, which’ll last 4 hours
User auth, is refreshable, implict auth you have to go round the loop
When you go round the loop you redirect a user To Twitch.
And the Twitch login cookie may have died or rendered invalid, which’ll mean the user is presented with the Twitch login screen instead of it just transparently making a new token.
LocalStorage lasting forever is irrelevant when compared to Twitch’s own Cookies/session.