if we are fully down, rest assured there will be no available hands to keep the status page updated

There is no quicker way for customers to lose trust in your service than it to be down and for them to not know that you're aware and trying to fix it as quickly as possible. One of the things Cloudflare gets right is the frequent public updates when there's a problem.

You should give someone the responsibility for keeping everyone up to date during an incident. It's a good idea to give that task to someone quite junior - they're not much help during the crisis, and they learn a lot about both the tech and communication by managing it.