The key is in your question
My feeling is that this is not how a just system would be designed - isn't the law clear enough that assuming everybody is informed of all the laws, trials should have the same outcome no matter who is involved?
The law system is very complex (and especially in the US, since US has a lot of very old laws, while many other countries just sit down and rewrite all laws after some major event like revolution). So knowing, understanding and being able to apply the laws requires a lot of knowledge, experience and discipline.
The laws themselves (phrasing in acts and codes) are one thing, whether and how they are being applied is another thing. Actually knowing the laws and being able to make use of them is very important for fair (or beneficial) outcomes. That's what lawyers are for.
Of course, a bad lawyer can be even more incompetent and useless than a determined citizen - as with any other specialists.
A hypothetical example. You buy a computer and it's malfunctioning and you file a lawsuit against the retailer. You come to the court and the defendant just doesn't come - he knows that according to an applicable law he can do that and noone will do anything about that because there's no evidence the subpoena has been served to him. You start panicking and now it's easier for the other party to talk you into a settlement on worse (for you of course) conditions. The defendant's lawyer acts according to the law - he just knows the procedure better than you.
So those media claims are especially right for large corporations. In many cases knowing the procedure and being able to apply the law is as important as the law itself. A qualified lawyer just knows the procedure better than average Joe does. A corporation will spend a huge sum an lawyers and they will use the procedure to literally exhaust the opponent.