The ethics of transgressions… what would you do?

True leadership involves courage and authenticity, so this is an opportunity I would take to model the 'speak up' cultural values of an organization. But I would sit on it for a while and then frame it carefully - the shortcomings have to be part of a larger story about how I reflected, changed and grew out of the honest feedback. Then the narrative is less about my shortcomings and more about how honest feedback and reflection turned this into a way to evolve and mature as a leader.

I don't know of many examples where this scenario is handled well - or made public. However the example of Dan Harmon, creator, writer and producer on Community - a popular TV show is one example of a senior leader owning previous transgressions and attempting to work with the harmed party when details of his transgressions were made public.
After a public apology he detailed the inappropriate behaviour and attempted to empathise with the transgressed party. In detailing the events, he highlighted that this transgression was not a single event - put a pathological pattern of behaviour and admitted that others would have been impacted too.
This approach is novel - he attempted to truth tell without sanitising the details, plus he limited the amount of excuses offered - after establishing his state of mind and personal history as context.
Institutional examples also highlight that the transgressed want acknowledgement of the events, often want an apology and if the events may make the transgressors role untenable the transgressor to resign or have their role terminated.
Obviously, this is a unique example, in some cases the person transgressed will not want details made public or may not be interested in an apology or demonstration of contrition from the transgressor.

No one expects leaders to be flawless. I think bringing it in to the open in a team environment and making an apology plus asking for feedback on future behaviour would be the way to go. That way the ambitious team members have an opportunity to learn from the experience.

There is various assumptions in this example. For instance there is an assumption that the feedback given is justified and fair.
Gaining feedback can be in various ways and some done better than other. For instance doing surveys that are basically not anonymous will not give frankness all the time.
Also to get no feedback is allowing people to carry on in their own path which could led to more negative outcomes.
Self reflection will be different for all of us. Those who spend the time and build the trust in my experience have the greatest potential to assist.
When we have stuffed up we should not do nothing. Depending on the issue some examples will need to be open (especially if impacting people) and others you just need to change your way of thinking and even ask someone close to monitor you or keep you accountable.