Judgment probably start from something like disciplinary committee, not exactly for dealing serious things like crimes.
Part of the education policy would also want the children to learn how to handle conflict on their own rather than calling adults for every single trouble (and grow up into adult who can't solve problem on their own, only deferring to authorities everytime).
Then as mentioned in 142.2, there's the Academy City's politics involved and they prefer to keep Anti Skill (adults) out of the matter.
Well, dirty adults won't stop at just solving issues, they will quick to think of how to profit from the situation.
Judgement is the individual disciplinary committees of the schools and thus their jurisdiction of actual law enforcement is supposed to be only the school grounds, dormitories, and student bodies their department is assigned to, unless they have permission from Antiskill, or they are in a suitation that Antiskill can't arrive to immediately. Shirai oversteps her duties constantly, and her department is centered on Uiharu and Saten's school, not Tokiwadai, which is often a point of confusion.
Because enforcing the law isn't really Judgement's job. At best they were supposed to be community watch type of deal or dealing with minor misdemeanors. It's just that deranged single target lesbian expanded her area of duty on her own. Probably in order to mirror oneeeeesama.
Anti-Skill is the organisation that is supposed to enforce the law and fight superpowered teenage twerps if need be.
Shirai's sense of justice isn't sourced from Misaka, just put into context by it and refined by it. Shirai's sense of justice seems so be sourced more a sense of noblesse oblige from her upbringing as a child of a wealthy family. The most important impact Misaka had on Shirai was make her less proud and self-righteous. In fact, her sense of justice usually is at its strongest when she's isolated from Misaka. It's brought up multiple times throughout the series that Misaka doesn't really need Shirai as an ally and vice versa, and Shirai is aware of this. For example, during the Sisters Arc she knew something was up but left it up to Kamijou instead, knowing she was incapable of preventing Misaka's self-destruction. Her relationship with Kamijou is interesting because she actually respects him quite a bit. Also in the few times they work together separate from Misaka, they get along fairly well because both of them possess a very mature and single minded sense of right and wrong which they act on without hesitation, and never compromise on no matter what. They also both possess a strong and resourceful intuition and clearheadedness in stressful situations, combined with a surprisingly strong ability to reflect on their decisions and are well aware of their limitations. Though the biggest difference between their respective justices is that Shirai's justice is a sense of noble selfless righteousness brought out of a sense of purpose and duty, while Kamijou's is fueled by his selfish and even greedy desire for self satisfaction in righting wrongs that upset him in the current moment (though I doubt Shirai is aware of this). When Shirai works with Misaka directly, she completely submits herself to Misaka's will, overriding her own values with Misaka's own more raw, emotional and reactive sense of justice. Misaka's sense of justice is peerless and unstoppable, but liable to shift suddenly due to her emotional instability. Misaka's more naturally kindhearted and altruistic than Shirai is, but this often makes Misaka susceptible to being manipulated or misguided. Shirai's got a heart of steel while Misaka's got a heart of a wrathful storm god. In terms of strength of will and character, Shirai's actually the stronger. She's fairly grounded and 'human' when compared to inhuman 'monsters' like Misaka and Kamijou.
TL;DR Shirai's actually a really compelling character when the story takes her seriously.