What is it that makes a leader? By the name, it is obviously someone that many can look up to, someone to guide those who follow through trouble. But what if they had troubles of their own? Is it acceptable for them to ask for help for their followers? Or will the look weak and useless?
Many 'natural' leaders have been paranoid for the longest time over this issue. If they make the slightest mistake, their followers will scoff at them and leave. Of course this is an exaggeration, seeing as arguably only the most perfect of society would do that. Every person is critical, we all have our opinions and it is up to us to either share them the way they are, or to look further into said topic. If a leader is wrong, then they are wrong, there is nothing they can do but move on.
So how would they move on? "Apologizing at such a position will weaken the leadership said person is carrying. Stick your head up and ignore your faults." Ignorance won't get you anywhere, and it most certainly won't get our fellow leader anywhere either. If they must look like a fool in order to apologize, then it is their duty to live up to this humiliation.
Followers are much more understanding then most leaders realize. They aren't mindless, they just have an inspiration. They'll gladly share their opinions; after all, no opinion is right. Facts can be incorrect too, so the odds of leader or follower being wrong on a certain subject are pretty high.
Leaders in general may be paranoid over the stress of having to make a good impression on those who look up to them. They aren't weak, and being wrong won't weaken them, it'll make them stronger. Apologizing for their mistakes is completely normal, and it is the right thing to do. Let the followers aid the leader, and let the leader show them a new way to success.