Neanderthal, I'm sorry you are feeling troubled about your relationship with your daughter. From what I have read and seen you are a very good father. You think about what she likes and needs, you interact and play with her, you care for her (even doing her hair!), you search for what is right and good for her, you want to set up your life so you can be the best father you can be for her. It's touching and wonderful.
That voice in your head telling you that you are a failure - that voice comes from somewhere deep in your past. That actually happened to you. At some point someone important told you that you are a failure and you believed it. You weren't a failure then. They failed you. You aren't a failure now. When that voice comes, think about little Neanderthal and tell him he is not a failure and try to give him what he needs to feel loved and secure. (sorry for the sappy armchair psychology, but I think it's true!)
Neanderthal, there are fathers who are failures. My father literally did not speak to me for four years (age 13-17) because he was so ashamed of me, ashamed that I was sexually active (in part because I didn't have anyone to go to after a 13yo rape and handled it by acting out sexually, and in part because he was so distant and I was desperate for male attention) and ashamed of the boyfriend I chose (who was black - I chose him because I was trying to be safe - he was best friends with the rapist). FOUR YEARS without one word! Not "how was your day?" Not "nice job with your SAT scores" (which were, btw, nearly perfect). Not "I'm proud of you for working 20 hours a week while maintaining perfect grades, the most demanding academic classes, varsity sports and a bunch of leadership positions." Not "please pass the salt." Literally, not ONE WORD! Then he died. He was lying on the gurney while the doctors were working on him telling them what was happening (he was a nurse so he knew that his chest was filling with fluid and he told them what was happening). And he STILL had not one word for me! Not I loved you the whole time and I'm sorry I was limited. Nothing! He was talking to the medics, as he was dying, knowing he was dying, and I got silence. Dude, you are NOT a bad father! You talk to her! You notice what she is interested in! You notice her accomplishments! You notice her sadness! If she was hurt she knows she could go to you! The ponytail is nothing, she herself told you that. She saw you trying to help her and that help being important to you, HER being important to you - THAT is what matters. You are not a perfect father but you are most definitely a good-enough father. When that voice comes that tells you that you are a failure, fight back with the truth.
And whatever scars you leave her with - and there will probably be some from you and from others - that's going to be what she has to work with through her life to find her strength and her resilience. And hopefully, those scars will leave vulnerabilities for someone who loves her to show their love. When I was processing all of that father-trauma, my husband said to me "With my dying breath I would tell you I love you" and Neanderthal, in that moment I was glad for every injury I ever suffered, so that he could say that to me and I could feel my need for him and the depths of his love.