But the jury doesn’t determine the sentence - only the verdict. So, government is still being given the power to kill.
And you think the jury members aren't aware that the sentence could be death?? I'm OK with it.
I believe that in most states the jury would convict on first degree murder. Then the jury goes into the penalty phase of the trial and after that part is over, they vote for either a life sentence or execution. That being the case, you are wrong. The jury does decide if he gets executed. The only part the government plays in this scenario is carrying out the sentence. I agree about one thing, many times they are doing a poor job of carrying out that sentence. Mummiah Jamal (whatever the heck his name is) is still on death row here in Pa. It's been something like 20 years now.
Also, potential jurors are asked before trial if they can convict someone of the death penalty. In fact, the Supreme Court just ruled on this very subject making it easier to toss potential jurors who do not properly answer questions about if they could convict someone of the death penatly. The case was Uttecht v. Brown.