For me pieces of infrastructure like Twitter shouldn't be owned by for profit companies. A Twitter-like public forum ought to be the town square of the web, and it should run along the same lines as Wikipedia. The data should be ours to do as we please with, there should be no purchasing access through adverts. App.net, and the insanely confusing concept of the Alpha app within it, might not have been the right solution, but they were asking the right question.
Unless you count the wave of innovation coming out of cryptocurrencies, which are really just protocols for managing a blockchain. A decentralised Twitter could theoretically be built with this model.
The only question is "who pays for the infrastructure". Wikipedia has done extremely well with its donation drives; that's very hard to repeat. App.net seems to have struggled to make users pay directly. Arguably the whatsapp charging model has worked best.