Here's the problem I see with Twitter... long time users are starting to bail out. The most recent example is hearing Marco Arment and the gang talk about doing exactly this on the latest ATP. The environment has gone from the initial 2006-era camaraderie to an ever more poisonous quagmire as more and more people have joined in. The problem becomes worse the more followers you get.
Twitter's model of open communication; where anyone following you can hear your voice, and anyone mentioning you will be heard by you, is both its strongest feature and its Achilles heel.
While the service was young and even today while you're getting started, this model is good. It allows a rapid growth of audience who are likely to be relevant to you. However, the longer things go on the more the balance tips to being overwhelmed by less valuable voices. The more likely you'll get abuse, ignorance, and stupidity flooding your timeline.
Twitter can fix this simply, and elegantly, by allowing its users to change the communication model. Copy Facebook.
At a certain point, it's not useful to constantly see what random other people throw at you by mentioning you. Twitter's existing 'Private Account' is not built to address this. Private accounts are for vetting your audience, stopping people from following you. It limits your ability to broadcast as well as your ability to hear from others.
Instead, offer us a 'Only people I follow appear in my mentions', or a one step removed 'only people I follow, and people they follow'.
Boom; the 'popular on twitter' tax becomes much more manageable.
Right now; this ability is only available when viewing your tweets from the website. I don't know anyone who uses Twitter via the website - we all use apps, on our phones or desktops.
UPDATE: turns out ATP mentioned this later in the episode. Verified accounts have this function. There's no easy way to get a verified account - those are still magical Twitter dictated things. That's not a solution.