She trained to protect herself and her mother, but my point is that she clearly is trying to be a superhero (Hawkeye), and most superheroes are vigilantes. That's the whole point of Captain America: Civil War, people taking the law into their own hands without authority. I'll concede she may not have intended to become a vigilante, but she is one now. She wears the costume of a famous vigilante and acts as a vigilante. Whether she planned to or not, she has engaged in vigilante activity.
As to the backstory presented in the credits, I still think it's unlikely that even the type and amount of training presented would enable a young woman that looks like Haley Steinfeld, or even a man, to be able to take on a group of male attackers. I'm having a harder time accepting these type of "one vs. many" battles than I used to, not just in this show, so I'm not just criticizing this instance. It's just not something that seems to be possible in real life. If Kate Bishop had superstrength and/or super healing it would help me accept her ability to stand up to a gang without major injury, but she is still a person of relatively normal strength, even if her some of her abilities are above normal. Her body does not look like one of a honed fighter, or even of an elite gymnast. I realize saying that the presence of superhuman abilities would make things more believable, but there's certain premises that I can allow for in the reality of a universe that has superpowers. A young woman that took (presumably) conventional martial arts and gymnastics just doesn't seem very likely to be able to fend off multiple attackers to me. If the show presented her training as a bit more intense and specialized, like with the Black Widow program, I'd be more inclined to believe it. I didn't get that from the animated intro or anything said in the show - like getting a black belt at a young age, as is stated in the show, may be unusual, but is not an indication of elite fighting ability. The belt system of commercial martial arts studios is known to be a cash grab that has little uniformity of ranking and ability among the different businesses that teach martial arts. The requirements for a black belt are different from school to school. The belt ranking system is simply a means of getting people to pay to progress through the individual system set by the business.
And I haven't forgotten Clint was a SHIELD agent, I'm saying they went to great length to show how famous he is, so it was weak how he walked past so many people without being recognized. Even if they didn't know he was Hawkeye (world famous superhero that doesn't wear a mask), none of the firefighters of that unit recognizes that he wasn't one of them. No police kept him from entering the crime scene. Sure, I admit that maybe it's possible for a person to go under police tape to enter a crime scene, walk up to a fire truck, open the door, put their bag inside and take a coat that's just laying there, grab a hat that's been left on the bumper, walk past law enforcement and fire fighters undetected and unquestioned then return to the truck to retrieve his bag. It's possible, but seems unlikely. Add in that he's incredibly famous and recognizable, and it gets harder for me to believe. That's all I'm saying. The writer(s) chose to make it easy for him to get in and out, and have a big, dumb clue right in his face as to who took his costume and where he would be because it advanced the plot quickly, and would put him in a funny situation later. Is it realistic? Debatable, but I don't feel it was, so I mentioned it because it stretched my suspension of disbelief. Same with the scene at the LARPing event. It didn't feel realistic in the reality they put Hawkeye in, so it took me out of said reality, reminded me that this is just a TV show.
Look. I know other people didn't have problems with these things, I'm just saying that I did, and it hurt my ability to enjoy the show so far. Maybe I'm nitpicking, fine. I just had issues and mentioned them here.
And since others mentioned the wine bottles not breaking, that's actually something I found realistic. Liquor bottles don't really shatter against people's heads like is normally shown in movies and TV shows. I've seen real-world footage of a liquor store robbery where one worker fought back and bloodied the robbers head by striking him multiple times with a bottle that never broke. Breaking a heavy glass bottle against a human head that will give way when struck is a lot harder to do than striking the edge of a bottle on a rigid, stationary surface, and even that can require a lot of force to do.