>>11165428She told Superman she doesn't want to use his symbol and she's still trying to figure out her own. The appeal of the character is how she always feels out of place and has to prove herself, that's why she overcompensates by being aggressive and assertive.
Between the lines, she understands the weight Superman's symbol carries so she doesn't want to wear it in case she fucks up and it makes her cousin look bad. She also sees Supergirl, a teenage version of herself everyone sees as the 'real' Supergirl, dressed in the red, yellow, and blue with the shield and needs to define herself as her own person, not just Superman's cousin.
She seems like a brute but is a very self-conscious and caring person trying to do the right thing.
Wearing an S shield means she's given up on finding her own path and become one of the half dozen clones, cousins, and children who dress like Superman, act like Superman, and never do anything wrong.
I say it again, the jacket is heresy.