I used to be an SRE at Atlassian in Sydney on a team that regularly dealt with high-severity incidents, and I was an incident manager for probably 5-10 high severity Jira cloud incidents during my tenure too, so perhaps I can give some insight. I left because the SRE org in general at the time was too reactionary, but their incident response process was quite mature (perhaps by necessity).
The first thing I'll say is that most incident responses are reasonably uneventful and very procedural. You do some initial digging to figure out scope if it's not immediately obvious, make sure service owners have been paged, create incident communication channels (at least a slack room if not a physical war room) and you pull people into it. The majority of the time spent by the incident manager is on internal and external comms to stakeholders, making sure everyone is working on something (and often more importantly that nobody is working on something you don't know about), and generally making sure nobody is blocked.
To be honest, despite the fact that it's more often dealing with complex systems for which there is a higher rate of change and the failure modes are often surprising, the general sentiment in a well-run incident war room resembles black box recordings of pilots during emergencies. Cool, calm, and collected. Everyone in these kinds of orgs tend to quickly learn that panic doesn't help, so people tend to be pretty chill in my experience. I work in finance now in an org with no formally defined incident response process and the difference is pretty stark in the incidents I've been exposed to, generally more chaotic as you describe.