Absolutely agreed, this is brilliant. Ideally partner with Patreon to bootstrap it, but allow anyone with a Twitter account to have patrons. (That also gets more people providing payment information, useful for other future purposes, such as making "sign in with Twitter" support "pay with Twitter".) Then start combining that with a Patreon-style "patron posts" mechanism, increase the limits on "patron posts", and perhaps later introduce a "circles"-like mechanism for non-patron accounts.
Imho Patreon isn't the right fit for Twitter. Twitter is about tiny impulses. Patreon is all about subscription-based stuff. Start with a tip jar of just "donate" - make only approved tweeters can host a tip jar for the first rollout. Make the TOS state tip jar cannot be used for conditional services, only for thanks for services freely provided.
Then expand to payment-locked content in tweets. "hey Johnathan Coulton posted a paywalled song click here to pay $1 to get it!". Make a platform where "1-click purchase" actually makes sense, since the patent's expiring anyways.
Giving users a direct financial incentive to game your system would probably turn your platform into a 'low-effort but popular' content mill in short order. (Imagine what would happen if people could withdraw reddit gold to cash.)
Drawing a line for 'quality control' becomes the next obvious step, but whenever money gets involved (especially other peoples'), any decision you make trying to decide for users how and for whom your system can be used ends up making it less viable to them, even if it's in the best interests of your platform or its quality.
"X with micropayments" isn't a new concept, and it's not a funding model panacea for a reason.
You can't game the system if you're getting paid by other users. We can argue about tweet quality, but it's no different than any other product or service that some people find valuable and others don't.
What you're describing is possible if Twitter agreed to pay top Tweeters to generate popular content, independent of user's volunteering to pay for that content. This would drive low-quality, low-effort content which nobody is interested in, but Twitter pays for anyways.
To keep scammers out they could start with restricting the feature to verified users. That would discourage beggars and hucksters until they work the kinks of admining and monitoring such a beast out.
