Why do they give chase in every episode? The student will come back to school the next day and could easily be arrested then. It's not the like the student can just move across the country.
There was actually one episode where a villain was indeed about to move across the country and was trying to steal a boatload of Smoiks (the fake money that candy and chip companies include in their product that you can trade for prizes) before he went to the middle of nowhere.
If they don't take care of it now, that means the student has all that time to attempt to make themselves not look guilty.
And sometimes the crime is on a time limit and the culprit can win or at least make everyone lose by playing for time.
Also, Rule of Cool. If every episode opted not to give chase and instead went "eh, we'll get him when he comes to school tomorrow," it would make for quite the boring series. Also keep in mind this entire show is basically one big parody of 70s buddy cop shows, which all pretty much had a chase in every episode as well.
Because, in a case of Reality Is Unrealistic, most criminals, even when cornered, will run. Even when there's many members of law enforcement after them, no safe hide away, no back up plan or no reason, humans have a fight or flight instinct, and thus every kid bolts for it because every instinct they have says it's what they should do. The Patrollers give chase because, like police in the real world, they have a duty to make sure the fleeing criminal doesn't harm someone or do more damage in the process of fleeing than they had by committing the original crime. Would you really expect the school's security to sit back and shrug as someone runs when said person has broken a dozen school rules already with no remorse? What kind of heroes would those people be?
Also, there were a few other episodes where there were genuine stakes involved, and the culprit getting away would have had consequences. In "The Shreds Fell Like Snowflakes", they needed to catch the culprit before they succeeded in keeping Vallejo from his re-election hearing, and in Masterstroke of Malevolence, the entire chase scene takes place in the precious few minutes remaining before the vandalism of the painting becomes permanent. There was also Nappers Never Sleep, where they had to get Duappy back from the culprit they were chasing before he starved to death.
Does it not bug anyone else that this series takes place in The School That Doesn't Do Anything? In every single chase scene the students are shown partaking in a seemingly infinite number of activities... except actually going to class. You see the kids do sports, arts and any club activity you can imagine, but you never see them behind a desk. Even the main characters seem to be at the Safety Patrol office the entire day. When do they actually, you know, get educated? It seems that all the effort Principal Folsom puts in maintaining the school's image makes her completely forget to make the school do what it's supposed to do: educate children. How is she not fired yet?
You seem to be watching this under the influence of the Tethercat Principle. Just because you don't see something doesn't mean it doesn't happen. The plots for the episodes are based around extracurricular activities and sports because they have the greatest potential for interesting plots and satire of school and cop show conventions. Not to mention there was an entire episode based around standardized testing.
Also the episode "A Cold Day at X" is essentially about Academic Integrity. Just because a lot of the action doesn't take place during school hours, doesn't mean kids aren't learning. It's also likely that they work during free periods.
Why the fuck is there an issue about diplomatic immunity when it comes to the trade of forged baseball cards and plates? The culprit is a student, not a diplomat! Now, if he committed a crime worthy of the police, then yes, he'd be in a lot of trouble.
Just to clarify, it's not all foreign students, just relatives of diplomats, which this kid was, as was presumably the previous one Fillmore busted.
The standardized testing episode raised protesting as a way for people to get their voices across, without breaking the rules or hurting others. All well and good, but doesn't that fall a bit flat in this case? Protest all they want, is anyone really going to listen to a bunch of kids on the matter? Not to mention the episode had one protester, and they were shown to be very ineffective and more of a joke.
In "The Currency of Doubt", Toby is set up by having his hard-boiled eggs replaced with cheese eggs to set off his lactose intolerance. However, he mentions that he eats hard-boiled eggs regularly, meaning he's familiar with their taste and texture, and when his lactose intolerance was brought up during the initial investigation, he pretty much says it's bad enough that he can't have lactose products at all. So how exactly did he miss the fact that the 4 eggs he managed to eat before he got sick were not only the wrong color (when Fillmore finds the remaining egg, it's uniformly yellow) but also were the wrong texture and didn't taste right, namely by tasting like something he had never eaten at all?