The way the Coruscant Guard and Commander Fox especially is portrayed in TCW fandom doesn't make sense.
In the first place, Fox's characterization is completely made up. I'm supposed to believe that the man who, while entering a room to arrest a crime lord, flips over the stairs is an exhausted and hyper-depressed? I'm supposed to believe the man who, when Ahsoka comes to see Letta, says "a lot of innocent people died in that blast. Good job capturing her." is bitter and hardened against the regular people of Coruscant? That the man who says "I can't say I blame you, Commander Tano. But all the same, you're under arrest." would easily break the law on his own justification? Well, I don't believe it, which makes almost all fic with Fox in it feel so OOC, often to the point I can't read them. This man is not amassing blackmail, he is not planning an assassination, and he isn't constantly interacting with Senators.
Moving on to the things generally wrong with how the Coruscant Guard is talked about:
They wouldn't be (and canonically aren't) stationed in the Senate building. Why? Because the Senate Guard is a thing that exists.
The Coruscant Guard (a) escort Senators and Jedi on diplomatic missions, (b) run security checkpoints, (c) investigate crimes relating to Republic security, and (d) guard the high security prison. Somehow their actual jobs barely ever talked about, despite the dact that they have the most unique skill sets in the GAR. Why are there no HCs about how those skills were developed? Did the Kaminoans have specialized classes or did they have to learn on the job?
Why is it always horrific abuse and never jurisdictional pissing contests? I promise, the Coruscant Security Force (CSF) and the Senate Guard have long standing issues over sharing responsibility for policing the Senate District, even before the Coruscant Guard is added to the mix. Plus Coruscant is an entire planet so there could easily be additional agencies.