Services that scales to multiple Discord servers use caching. It’s highly unlikely that every single server the bot is in is wanting to get notifications for unique channels that no other Discord server has requested. So if hundreds of channels want to know when the TwitchDev channel is live, it’s still only going to make 1 request and cache the result, not 1 request for each server that wants notifications about TwitchDev going live.
Some services do have an increase rate limit for the API, but I don’t think the form to request an increased is used any more, and for the sort of service you’re looking to make you shouldn’t need an increased rate limit because the appropriate way to go about it would be to have users auth with your app and you simply use their token so you could scale to any size and wouldn’t even come close to the default rate limit on each token.