Coming from someone who has a scar on their face as well it really depends on how you got it and how big it is. Like in Yuma's case he got it by 'saving' Shizuya so he may feel proud of it. Other people may just not care what others think. In my case I just laugh at how stupid I was being whenever I see it. (I am in no way speaking for other people)
I see you make good points. I happen to have scars somewhere else and they bothered me at first but as time went on I got better about how I looked at them. I changed my view point. I was just like wow he got there quicker then I did （￣へ￣）
Some people require others to help them embrace their scar. My dad just kept making jokes about what actually happened and then at one point I was actually okay with it(⌒▽⌒)
The way he is drawn feels like he is inflatable in comparison to everyone else. He stands out too much I feel. I also cant fathom being ok with getting a scar on your face that easily.