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Why don't they authenticate with their own Twitter account? I'm not sure why they would need mine.

The Twitter API calls are rate limited and it would be impossible to handle any meaningful traffic.

Feels like click bait to me. They could use their own authentication token to use the Twitter API. Not mine.

Click bait in what sense? They are only requesting access to read (not post) tweets.

And the reason is because that's how Twitter sort of forces you to do it with their API rate limiting.