Ideally yes, but is it a hard-enough responsibility to mandate? If not, it seems like there is a financial opportunity for a company to fill that request. Especially in the case of time-sensitive emergencies, it seems like "informing the public as things happen" isn't the number one priority.

CIP: Living in downtown Portland, I have way better luck searching recent "#portlandprotests" tweets than I do checking the PPB Twitter feed. The latter usually shows up within 24 hours, but doesn't really solve the immediate curiosity of "why is there a helicopter hovering above my house right now".