People like to see character growth, boring stories don't usually have this. I watched a movie a few years ago where it was just the MC going around killing people for 2 hours and then he got to the main bad enemy and he fought him and won and the end.
There is nothing wrong with writing about a skilled person, but they can't just auto win. As you said, some times they do fail. This is okay! Failing doesn't mean they are incompetent. Failing simply means that their current idea did not work and they need to rethink.
You mentioned that the MC is a hothead! Really drive home that weakness if that's the one weakness you want to create. Show how it gets her into bad situations because of her temper, but then as the story progresses, have her learn how to handle it. Maybe not get rid of it, but learn how to handle it so she is able to think more rationally and mature. Possibly have one of her failures even be due to her being a hothead. Have that failure be major enough that she might want to consider holding her tongue or what ever the situation would be.
Hiding their past isn't a weakness. It creates a mystery about a character that should eventually get revealed slowly through the story.
Most people who are extremely smart also tend to lack common sense. Most people who have amazing street smarts, may not be the best people when it comes to academics. Stereotyped? Yes, but generally that is how it plays out. If your character is smart, plays an instrument, can build a rocket in 3 days, is able to run for 20 miles, then balance it out. Make it so her hotheadedness plays a factor that because she is so smart, she comes off arrogant and is easily angered by others and thinks she knows everything. When she gets advice about being street smart, refusing to listen to other people's advice which ends up causing her to get into trouble and she refuses to listen until something major happened (say she was kidnapped or raped or what ever the story is), and then she starts being more humble, and not thinking she knows everything about everything.
In the end, as long as you don't make the MC a god character that wins every time, and does everything correctly, you are okay. There are real life people who are generally good at many areas in life so why should characters be any different? Just make sure that on the few weaknesses you provide, that they balance out her good qualities. Or, go the other way, make it so her hotheadedness leads to her demise and ultimately causes her to lose everything.