It's not always so simple. Taking blame of others' mistakes can come across like regret for hiring or trusting that person. Sometimes part of giving people responsibility is letting them deal with the mistakes they make.
Outside of the team, ultimately the manager is responsible for delivering. That is their job, and with a good manager, the buck stops at them - without exception. If they don't deliver, it is their failure - even if that failure is a result of their reports failing to deliver, or unrealistic requirements from their superiors. Perhaps the appropriate internal action for that failure is to discipline their reports, but ultimately they are the captain of their ship, and should be the first one on and the last one off.
Obviously, all this assumes that it actually was a valid mistake. In the rare case where the failure was caused by gross negligence or intentional misconduct, then you should take responsibility for hiring and trusting them, and then you should fire them.