Is self-criticism ever "real"?

The question asked by @janbb got me thinking…:
Most people have what they consider “faults”. For example, I think I’m too controlling – I tend to feel responsible for things I can’t do anything about. Also, I have a hard time being ‘nice’. When I’m unhappy, even the people I love get on my nerves and during those times, I think I’m pretty hard to deal with. In theory, I would be able to change that, I suppose. I could be more punctual, less controlling…friendlier…but if I could magically change, I don’t know that I would. I’d be someone different, and I wouldn’t like that. Turns out I’m not really criticizing myself then, right? What about you? Would you really like to change some things you don’t like about yourself? Are you actively trying to? Or do you acknowledge your faults, but live with them?
I’m beginning to wonder whether what I should be trying to change is my lack of self-criticism…