Earlier quoted context omitted.
To me, the role you are describing is a principal engineer or team lead, not a manager. Simply not true however that a good manager can't lead the team in a high pressure situation. I'd say that exactly what a good manager could do well. Obviously they won't be making overtly technical decisions, that's what you are for. They can make business decisions, provide cover, get resources, communicate to other stakeholders…
I don’t see much value added by a manager that you described. What you are describing is more of a product manager role.
I wouldn't call what I'm describing as product management, although it's possible they could do general management too.