I don't know the specifics about Twitter's API saga over the years but... why isn't this the case? Why does Twitter need to be involved on the client side with consumption of the API?
Twitter is an advertising company. Alternative clients do not show Twitter's ads nor return to Twitter user data, so Twitter makes no money when alternative clients get used but still incurs bandwidth costs. There is no profit, only cost, in providing an exchange for short text messages; the profit all comes in the advertising, and that requires control of the client UI.
They don't need to be involved, they want to be involved, and they have the legal backing to do so.
I'm not sure how this would be any different than other software projects that use an API they don't own. Things like Cider for Apple Music, Ripcord for Slack, and the many Spotify clients all come to mind. Perhaps those are less aggressive in pursing 3rd party clients?
The issue is why can't Twitter charge for tweets via their API, but offer them for free in a browser if they are served alongside ads. Perhaps their own internal research suggests so few people would be willing to pay that it wouldn't be worth maintaining the API, but it would be nice to hear more info about this.