These indices are officially published once a day, so technically JPMorgan doesn't have to publish real-time data of the index. There are no exchange-traded product on top of this, so if JPMorgan sells you a swap that settles on some value of the index, they also don't need to provide a live value for trading.

Also the computation needs to be transparent and reproducible. I didn't read the spec, but I doubt that this is a complex NLP model.

Technically, some people are already using super fast twitter api for sentiment analysis, so they probably do it the way you say, but they're on the buy-side, not the bank side.

EDIT: basically the purpose of this index is to hedge your vol exposure to Trump's tweets. JPMorgan will sell you a swap that will pay more or less depending on what Trump says. They'll have a fairly large bid/ask spread cause this is in fact quite random, but statistically, they found that this product may make sense for some people.