You should only be using Twitch.api if user is logged in. Unauthenticated requests with the JS SDK fail, as it does not send the client ID with its API requests, only the user’s oauth. This is due to being from a time when jsonp had to be used (headers impossible) and the client_id query parameter had a different use on the streams endpoint.
For follower alerts to be used with browser source in OBS, you’re probably better off with basic $.ajax from jquery, or possibly even XHR or fetch.