Sometimes communicating is worth it for interrupting the teamPerhaps, but the first thing any emergency responder will tell you is that there should be one incident commander, and that all communication and allocation of resources should go through that one person. And that the person with the incident commander role should not carry out any operational duties besides coordinate and control, because those other tasks require the same mental faculties, but use them in a different way, making them less effective in either.
If there is a requirement to update customers affected by an incident, the company should have processes in place to streamline the flow of information without interrupting the team. And the team responsible for updating customers should only communicate with the incident commander, and absolutely steer clear of the operational responders. If specific details are required, it falls to the incident commander to gather those details from the team, not to outside people running interference.