Otherwise, it's mostly things you can do before and after an incident like this.
Depending on your role and place in hierarchy, yes DO sit on calls, even if you are quiet; to understand what's going on, to help prioritize if needed, to see if your experienced "spidey sense" catches some risk others may not even though they're closer to the problem (or precisely because of it), encourage, guide, and focus; but all of that without being overbearing or making them feel watched. You need to feel like an ally, part of the team, a friend as opposed to political officer. This again is best achieved ahead of time by demonstrating your value to the team by being supportive and effective always.
The main thing you can do is after all is over to make people feel awesome, give them recognition in all the ways possible: bonus, extra vacation days to compensate for the crunch time, etc
But if you still have your technical chops - jump in the trenches and debug!
In your experience, when something like this happens, what can leaders do to help in the moment? I'm an engineering manager, and watching people getting over-stressed is much easier to do when you're physically in the office. Working remotely, it's easy to slip in the trap of either assuming no news is good news, or the opposite approach of asking "are you OK?" every 2 minutes. The only solution I've found is to all sit on a call, and pay attention to people's tone of voice, but I don't know whether that actually contributes to people's stress levels.