People have argued the right to not incriminate themselves right up to the European courts, but it was rejected. When you are arrested in the UK you are told that if you fail to mention when questioned anything you later rely on in court it may harm your defence, so there is no right to silence either.
I enquired about that. Here's a situation where it "may harm your defence". Let's say you are a crime suspect. You say you have a witness you can give evidence that you were nowhere near the crime scene. What should happen is that you tell the police who your witness is, they question your witness, and either let you go because the witness convinces them you were not there, or the witness says your story is bullshit, or they decide not to believe your witness and investigate further. In the last two cases this will then come up in your court case.
But let's say you refuse to say who the witness is. The police has no chance to check if the witness says the truth or not because they don't know who the witness is. And in your court case you suddenly present the witness. In that case, the judge can throw that witness out because the police had no chance to investigate.
So you do have the right to remain silence, and it doesn't harm your defense if you remain silent all the time including the court case. It _can_ harm your defense if you remain silent but only until you appear in court.