Mandatory promoted tweets for 3rd party clients maybe? I use a 3rd party android twitter client due to its superior UX compared to the official app, but I wouldn't mind getting served promoted tweets in that client at all.
I am not sure it would work because I don't believe plain ol' tweets work as an advertising mechanism.
Hence Twitter getting more and more creative with formats: app installs, cards, videos, and have you seen the new "send me spam" ads? http://i.imgur.com/h4H7RQw.png
They can do that, and they can also charge the app suppliers per customer if they don't show the ads. I'd willingly pay to have no ads, better functionality etc.
No they don't. They had $1.4 billion in revenue, so they must have to have profits lower than that. In fact they lost half a billion dollars last year.