Okay, so. Daredevil's inciting incident, at least in some versions, is hearing a little girl being abused at night by her father.
Matt could only hear this because he is enhanced. He lives in a world where the rights of enhanced individuals are extremely dicey. There are many organizations which love to snap up enhanced individuals and use them for their own purposes, which Matt knows very well, because he narrowly avoided being made a child soldier by one of them.
So trying to submit evidence collected via "having powers" is not a viable option for him. What can he do? Well, he can follow all appropriate channels to report the abuse.
Here is where it is going to get real for a minute. Warning for discussion of child abuse.
By coincidence, I have been in this exact situation. Little girl an apartment over, I could hear the abuse through the walls. So I will tell you what happens. You call child protective services. You file a complaint. You do this over and over, as new incidents arise. Perhaps police come. If they do, they knock on the apartment in question, and when no one answers, they shrug and leave. You offer to let them into your apartment so that they can hear the abuse - won't that give them probable cause to do something, anything? - but they refuse.
You continue filing complaints. You do this until you move apartments. To the best of your knowledge nobody ever helps that little girl.
Now imagine that you can hear more. You can hear, in great detail, a little girl being abused. You have gone through every legal channel you can. You understand the system very well because you are literally studying to be a lawyer. You understand, clearly, that there is nothing else the system is able to do. This girl is on her own.
Except that there is you. You have your own, deep rooted reason to avoid fighting. But you also have that thirst for justice that is the entire reason you're becoming a lawyer in the first place. To help people. And you can't stand to listen to a child being assaulted, night after night, while the police brush you off and nothing gets done. So you put the abuser in the hospital.
There is no contradiction. Matt Murdock believes in justice. That's why he's a lawyer. There's so much he can do inside the system to create lasting, positive change. And he can even potentially influence changes to the system - think of landmark cases. Matt being a lawyer, being a good lawyer is a BIG deal in the marvel universe, it means he can work on an axis to protect people that other heroes can't (shout out to She-Hulk, the only other lawyer hero i can think of besides DD. Great minds think alike.) And when the system falls short, that's where Daredevil steps in. And not only that, but because of DD's powers, he is uniquely good at being a vigilante.
In most every case, especially because he usually limits himself to Hell's Kitchen, Matt comes in with a ton of context to any given situation. It's like he's constantly running a low level spying operation. He knows his enemies far better than they know him. This guy can eat his lunch outside your apartment building while you're planning your next crime and just like that you've been made and don't even know it. Matt is a very good vigilante because he has a very good insight into people. The ability to know when someone is being truthful is worth its weight in gold.
Why would you look at all that and think he has a guilt complex? There's nothing contradictory about Matt Murdock, lawyer and vigilante. Those are two halves of a whole person who is all about justice. He believes in the system, and uses it to the fullest. He sees where it falls short and fills in those gaps with his own fists. Matt is not some twisted up Catholic martyr, he is an extremely intelligent man without fear dedicated to justice and with the wisdom to know that justice and the law of the land are not always the same thing.
There. That's my rambly manifesto. I hope it's legible. I'll edit if i remember later. Frank Miller if I see you it's on sight.