I think you underestimate how much pressure you can apply to people, from an organization who has been using leverage and pressure as their primary tools.
As a parallel, lets say you have a friend who grows a little weed. The police come to you and want you to get some pictures of your friend's grow operation so they can raid him and arrest him. No one will ever know you did it. You refuse to do so because you're no snitch.
Every day when you go to work you start getting pulled over for minor traffic issues and given outrageous tickets. You're having to go to traffic court constantly and deal with it. One day an officer reaches in and throws in a bag of something into your car, "Good thing this is only oregano this time" and tickets you again for going 60 in a 45. Your girlfriend starts getting the same treatment as you are. When you go to work your boss wants to know why the police had a meeting with him about you and possible drug issues. A week later you're pulled over and arrested because you didn't show up for a jury summons that somehow went missing from your mailbox and now there's a bench warrant for failure to appear.
Exactly how much shit are you going to put up with for something that will have zero effect on you?
Now imagine you're the head of a giant world wide company with billions in revenues. You've busted your ass to get where you are despite the hardships. Your position can be easily lost due to any issues regarding company performance. Very serious people from alphabet soup agencies come to you and explain that due to national security they need to be able to hack your product to prevent the death of American lives. Also, you can't even discuss this with your attorney because, you guessed it, national security and we say so.
But don't worry, we're going to have the FBI come to you publically the next time there's an "incident" and you can make your refusals then, publically. Right now however, we need the backdoors so we can stop the next 9/11. Google and Yahoo already let the NSA read the emails of their customers. You don't want to be responsible for the next terrorist incident do you?