Yeah I built an integration like this for a hackathon at work (we did call center software, e.g. we serve the people that serve people). The system categorized tweets so that the customer could respond to issues and such - the obvious value add is it's unlikely a customer will come to you directly, but they'd likely complain online.

Somewhat related I've noticed that some of the really successful open source projects meet people where they are and are really accessible via channels like Twitter (SpringBoot and Auth0 are good examples of this, from my experience).