One lesson I learned is that integrity itself isn't objective, at all. Everyone thinks they have high integrity, and it's easy to rationalize another person's actions as stemming from integrity if they are acting in your interests. When it comes down to it, as you state, you can't really grok a person until you are in a crisis. Similarly, it's hard to grok your
own beliefs about what makes for integrity until such a crisis.
For example, a company usually has a mission. At the same time, up to a point a CEO has a duty to his/her employees' well being. In a crisis, these may be in conflict. And in a crisis it can be very hard to tell if a decision cutting one way or the other is stemming from expediency, a conflict of interest, on-the-job evolution of a person's values, or an underlying consistent principled view that is now manifesting in action. (The last of which is what one would arguably point at as the person demonstrating integrity.)