What would a just legal system even look like in a socialist society? It seems like the legal system under capitalism still holds on to a lot of weird feudal stuff, so it’s kind of hard for me to get my head around what would come after this, if we completely destroyed the bourgeois legal system.
I have no education in law and it all seems intentionally opaque, so I feel like I can’t even begin to imagine an alternative.
Common law itself is fine -- it's just a way of saying "we should all agree to judge people the same way" -- but there is no effort (read: no money) to try and simplify it. So it ends up being a clusterfuck, especially now that most case decisions are available online.
The short answer to your question is: Corporate Capital interests prefer a system in which they can move the law by influencing Judges -- mainly because all of the Judges are former partners at the law firms that represented said Corporate Capital.
But the fix, once you remove those influences, can be as simple as funding and/or legitimizing certain existing efforts to take the common law and simplify it into a usable "code" of rules.
A good example if you want to learn more is with the various Restatements. They take a close look at everything that's been decided and put together a more simplified and organized understanding of the state of the law. A lot of judges and lawyers will rely on these heavily, almost as if they were law. But they're not the law and it's an important distinction where the money really matters.
If you nationalized that industry and funded it, removed the corporate lobbying interests above, and passed legislation making the State Restatements actual law, it would go a long way towards having a better legal system and with relatively minor changes.