Without API and without web access, where are you going to read the tweets from?
The only option left is unofficial API stubs, which essentially scrape the web requests, but those also will require a login, so the high activity will be easy to spot and block by Twitter.
Unless they have a whole network of accounts. That'd work but it's work to do.
There's a nitter fork (in the sense of 'PRs not merged yet') that uses the admin's burner accounts to retrieve tweets, and it still works. But I think it hits rate limits fairly often.
edit: Well, worked. Seems down right now, possibly because it was the only instance that still worked.
https://nitter.net/ https://nitter.net/about
See an example: https://nitter.net/elonmusk