That wont help them at all with a website like they said they are using, and there’s no need for that scope just to check if a channel is live, and using the search channels endpoint is also not the ideal way to check if a stream is live or not.
They can’t include an OAuth token on the page itself as it’s a violation of the dev agreement to make OAuth tokens public, and it would require updating the website each time the token expires. The correct way to do this client-side, such as on a website, would be to do the implict auth flow and have the user login to the site, then grab the token from the URL hash and use the Get Streams endpoint https://dev.twitch.tv/docs/api/reference#get-streams