It takes a strong person to admit when they are wrong, and even stronger person to lead others through their place of vulnerability. A leader who recognizes their mistakes and provides genuine apologize for it will create a culture of authenticity, humanity, relatedness, and acceptance. We all make mistakes, and we naturally seek forgiveness for them. A leader who can seek it, will have the capacity to give it...this is empathy.
A leader who refuses to admit mistakes creates a fearful team who will never show themselves as whole. This will be a team who builds walls of protection around themselves. They will become "yes" people, often not proactively communicating, productivity is hard to articulate, and engagement is low when interactions are made. These team members are often reference their time as 'just doing what they have to do to get by'.
To truly know someone, means you know the whole person, and since we all make mistakes, it means knowing the mistakes of a person; it is the only way trust is developed. A leader who fails to admit mistakes is characterized as prioritizing appearances over purpose, which translates to "fake". Fake will never be trusted.
A leader who always has to be right, creates a team who can never admit failure, and is a team who can never be whole.