The problem with justice is determining who deserves it. Measuring culpability is no simple science.
If we're talking about murder, then let's consider motive. Who is more culpable: one who kills in cold blood or one who kills in passion? Let's be more specific: murderer A robs a bank, and during the robbery, shoots the teller; murderer B is a law-abiding citizen whose daughter was raped by a depraved individual, and in a moment of passion, he hunts down and kills the rapist. Both committed murder, but who deserves justice?
I think as a society we would agree that while both murderers are responsible for their actions, murderer B is less responsible than murderer A, as his emotional state, induced by a signifantly emotional and personal event, led to a crime of passion rather than murderer A's act of cold blood, and that murderer B is much less depraved than murderer A. As such, we would apply a significantly lower punishment on murderer B than on murderer A.
Now, no matter what the circumstances, murderer A's going down. But let's see how this plays out sans the 5th for murderer B. Without the Fifth Amendment, one of two things happen: Either he/she lies about committing the murder, or he/she tells the truth about committing the murder. And here's where the fifth amentment makes the difference...
Say murderer B lies about the killing, and is caught doing so. The act of committing a lie will very likely prejudice the measurement of their culpability by the judge or jury. (The human thought process would be something along the lines of: "If he's capable of lying under oath, what else is he capable of?") This would negatively impacting the sentence given.
On the other hand, say murderer B admits his guilt. Then there's no need for a prosecutor to measure culpability. Why does the state need to know why it happened, when it has an admission of guilt served up on a silver platter? The motive for the crime is now irrelevant and moot. While a plea deal might be worked out to reduce the sentence, murderer B will get stuck with likely the same punishment as murderer A. This then also negatively impacts the sentence given.
Pleading the fifth forces the state to carry out a trial, find facts, analyze them, and deliberate on them. This will provide a much more accurate measurement of culpability, allowing the state to offer murderer B a more appropriate punishment to best fit the individual's crimes.