To get started, I'm not sure if everything I address falls under the responsibilities of the judiciary but I hope everything falls on the right ears in the end. Also I will not go into detail with everything right now.
About the fundamentals, I'm pretty sure only way to have a properly functioning legal system is some kind of OOC plea after every arrest. It's simply necessary to not overflow District Attorney's Office with having to arraign everyone even if they have no plan to fight the charges.
OOCly pleading not guilty means that District Attorney's Office will start the procedure to arraign the defendant. To avoid situation where one waits for arraignment in jail longer than he would serve if convicted, decent bail system needs to be implemented.
OOCly pleading guilty should be roleplayed as if defendant was taken to the NPC judge where he pleaded guilty. Then he shall be imprisoned by the LEO in accordance with standards set in the penal code.
Additionally some cases should be forced to the court. (CKs, some strike system for repeated offenders etc)
Difficult part if finding the right environment for trials. I encourage you to keep it in game as much as possible, however I know that there are issues with matching timezones etc. It would need discussion in larger group.
There are also few more specific points I'd like to bring out.
1) Introducing court clerk as an actual player filled position would not be smart move. At least not right after the launch as we do not know the actual interest in the judiciary roleplay, it would be somewhat waste of resource. You could simply NPC court clerk as it holds no crucial functions (If there are some, feel free to correct me).
2) Warrants should be a must. I have explained it's importance before so I will simply copy it. Warrant applications can be distributed through forum or even Discord with proper plugin.
3) Straightforward legislation for procedural aspects should be set in stone. It may not seem that important at first but it can make the system collapse in long term. Also adjusting some of the Californian landmark cases to San Andreas will help create the realistic environment.
4) Introducing state level RICO act would add depth to both legal and illegal roleplay. Plus, possibility of your illegally gained assets being confiscated, brings in the need for white collar aspect in organised crime. Roleplaying around legal fronts would then serve it's purpose as it allows you to cover the tracks of illegally obtained cash.