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.
Within the team, I think that is entirely appropriate. People thrive on, and should be given responsibility. Individuals should be rewarded or disciplined appropriately. That is part of the manager's job.
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.
In this framework, you don't take the blame for others' mistakes. You take the blame for your team's mistakes, and take responsibility for fixing them. You explicitly do not place blame on the person who actually made the mistake, nor do you imply that you should not have hired or trusted them.
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.
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.