To be fair: I want the people who can speak most competently to a problem to be working on it versus writing commentary. Those highly-technical people can shine in the postmortem report.

I disagree; the communication aspects - including external - are a hugely important part of any incident response. IME the biggest challenges in these situations are the many levels & combinations of communication and coordination. You don't need your best DevOps person doing it, but there should be plenty of competent technical product managers or engineering managers who could do this well.

Microsoft is not a small startup where the CTO is getting pulled into firefighting.

They should have established procedures for large-scale events, and they must include a coordinator whose entire job is to manage the external and internal communication.