"Did a bomb go off in our datacenter? No? It can wait then."
While some people -- healthily or not -- are willing to bust their butts (with or without extra compensation) when they think everyone is in it together, it will generally kill morale and willingness to do such when you realize when someone else 'doesn't have time', the deadline is moved, it's only you expected to work evenings and weekends and extra hours to meet the deadline. Why is your work-life balance less important than theirs exactly? (Only because they can get away with it...)
And while not always as stark as your example, people are often shown this all the time, that not everyone's inconvenience is in fact treated equally, that the "deadline that can't be missed" really only applies to some people, and is moved for others.
I do think it can be ok to work extra hours to seize an unusual opportunity or deal with some truly unexpected problem. (E.g., the destruction of the World Trade Center in 2001 required a lot of tech people to scramble.) But that has to be balanced by something good for the workers and painful for the business, like an equivalent amount time off or significant overtime pay.
But workers shouldn't have to spend their spare time making up for managerial incompetence and malpractice.