Yes, I don't understand that either. A power user of Twitter needs a suite of third-party apps and pays for them. Why not provide them for premium users? This makes Twitter officially pay-to-win, but it already is inofficially.

Features I use via third parties is mostly analytics. When to schedule tweets? Whom to follow/unfollow? Which accounts are fake/spam? Who favs/retweets me a lot? I would like to see some overlay in the UI which gives me stats about other users.

If I would have something to sell, then we enter the world of conversion tracking. Twitter should support A/B testing. The interface for teams managing a single account seems weak (haven't used that much).

Imho the core service of Twitter is to connect brands with people. It still does that job better than Facebook.