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.
You all are right that a jury makes the decision whether to convict. But based on what? Who is responsible for conducting a fair trial? Many DA’s are not above bending the rules to win, and many judges don’t stop them. Furthermore, the jurors can still be wrong. Ever hear of a “stacked jury”? Bottom line is, there are far too many things that can go wrong, and there’s far too much trust put in the state to expect trials to be conducted flawlessly. Our legal system is not 100%. The death penalty is. Until our legal system is perfect (i.e. never), we should not use the death penalty.