Is engaging with the site just reading its content, not liking or retweeting? Sometimes I just want to read a tweet thread that was sent to me without ditractions in a straightforward UI.
1. do not charge anything 2. the original publisher did not charge anything 3. Your presentation is less user-hostile in the sense of reduced TOS, easier sign up, cleaner visual presentation, etc
Then you should be able to repro it and cannot be sued for copyright infringement.