The data that twitter can supply is good for basic mentions, the basic smell of the situation - just not it's meaning -good/ bad, action required, not required. It's a canary in the mine, but it can't tell you what gas it is. This what many brand/ marcomms research dashboards supply - but many aren't optimised for the 'now take action' moment - they are mostly like webcams of rivers - an ongoing situation goes on.
What does work is smart , linguistics trained researchers, alerted by flagged moments in data inputs, then raking through data/ tweets to find meaning. Machines are useful for sorting these out but real meaning comes from human analysis still. Your engineers could help by writing some scripts that help the sifting/ alerting.
Everywhere data is used for analysis, a human is required for the real interpretation. The robots just aren't that smart yet.