For some reason you are a suspect in a case, say murder. Now this murder you did not commit, but there is evidence that makes you a suspect, and the court is willing to try the case based on the limited evidence against you. Even if the evidence is failing to convice the jury of your guilt, they may try to put you on the stand to testify as to where you were on that night.
Perhaps you were engaged in some other crime (however minor), you now are compelled to testify and confess, and risk being charged with that crime. Doing so possibly destroys your credibility with the humans (who are automatically fallible) on the jury (remember, they may just go "oh he's a criminal anyway, he probably did this too") and secures a confession for whatever else you did. If you retain the right to remain silent, but it is allowed to be used against you: You've still destroyed your credibility and the jury can now weigh this while deciding your guilt or innocence.
Also, your FAIL3 doesn't really apply. Self-incrimination automatically assumes you have done something that would violate a law, but there is not enough/any outside proof. Possibly an unjust law.
If we had a perfect system where we only charged criminals, tried by an infallible judge/jury.. we wouldn't need the 4th or 5th amendments.