To make it possible, it'd have to be a MVP - just basic twitter functionality with all posts stored in memory and accessible for only a few days. Just basic, high-performance, low-latency micro-blogging with an extendable API. Everything else would be outsourced to the community: search, direct messaging, analytics, history, advanced functionality, etc.
It would be like a centralized mastodon, but with a b-corporation type structure.
However it’s very hard to make a hit social media even when there is a market (see: decent dating apps), so i wouldn’t be optimistic.
That being said, it seems like a fun project, and if you use existing technologies you can probably make a prototype/MVP very quickly