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... All the bits that need doing but would be a huge hassle for the techies who are trying to fix the issue.
Not everything of value is technical. That you don't see the value is either because you have great managers shielding from having to deal with all that, or you have always worked in a place that combines management and technical responsibilities (which I never have except for very small companies).
I wouldn't call what I'm describing as product management, although it's possible they could do general management too.
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... All the bits that need doing but would be a huge hassle for the techies who are trying to fix the issue.