Yes, mistakes can be very hard to live down. I’m glad that you identify your perception as a feeling rather than a fact. If you are famous and you make a mistake, it really could follow you for many years, maybe even for life.
If instead, you interact with a relatively small group of people on a regular basis, it’s amazing how quickly you can “overwrite” how others perceive you. When a famous person makes a mistake, those who hear about it form a “snapshot image” of who that person is. Then usually, they hear nothing more about that person, so that snapshot remains intact and that person is forever identified with whatever mistakes they made.
If your circle of friends and associates is much smaller, and you interact with them on a more regular basis, they gain a far more complete picture of who you are and to what extent you learn from your mistakes. They also come to know how reliable you are. As all this happens, past mistakes fade into distant memory and surprisingly often, at least functionally, are forgotten.
You are no longer the person you were when the mistake was made. Having acknowledged the mistake, you are less likely to make a similar mistake than someone who has never made such a mistake. A person can choose to let mistakes become instrumental in gaining wisdom and strength.
What would you say to a person who made a similar mistake and is now trying to move beyond it?
There are some people who take pains to remember the mistakes of others, in an attempt to gain psychological advantage and even dominance. I don’t think there’s much to be done about that, other than to steer clear of them as much as possible.
Those around you who are decent, are much more concerned about the person that you have reliably proven to be and will perceive you based on who you have been in your recent past.
Everyone makes mistakes. If they aren’t on record, it’s because they weren’t caught. Yes, many of those who have not been caught look down their noses at those who have. It would be a mistake to resent them for it because such resentment interferes with one’s own emotional well-being.