But you haven't been able to establish beyond doubt. Not even close. In fact your proposal was to abandon the jury and let an expert decide whether a plea of diminished responsibility should be permitted. That creates considerably more doubt than even the current system.
And diminished responibilty isn't even the only factor at play. What about coercion, blackmail, police fit ups, or simple insanity which can still be used as a defence to a crime. These things may not be very likely, but not very likely is not enough for beyond doubt. Your toytown understanding of the legal system, where only the event matters and not the circumstances leading to it, is a recipe for injustice. You also haven't explained who decides, and how, the parameters for beyond doubt are established. How fo you pick, beyond doubt, which cases justify a capital trial?
At least be honest and acknowledge that beyond doubt is impossible. That no matter what safeguards there will always be injustices. Innocent people will die. Information that might have become available in the future will be lost. Juries will be biased because there's really no way round the death qualification in jury selection. The power of the state will be massively increased. Witnesses may be less willing to testify if it means potentially a death on their conscience. Many families of victims will oppose it. Judges and prosecutors opposed to the death penalty might leave the legal system. Everyone involved in the process of execution from the jury to the executioner is likely to be tainted and in some way traumatised by the process. And for what? To satisfy some vague demand for vengeance shared some of the population or to satisfy your personal sense of morality because locking someone up in a cage for the rest of their life is not enough for you.