When discord loads, it copies it out of localstorage into a js var and deletes the localstorage. So if you examine localstorage it will be empty. On page unload it copies it back into localstorage. So if you want to see the value, you have to make sure no discord tabs are open in your browser.

I believe one reason for this is to prevent self-xss. It's hard for a malicious person to write a snippet of js to steal the login cookie now that it's no longer in localstorage. Another reason might be to prevent bots. It's hard for someone to automate a discord account if there's no way to get the account's login cookie.