Go ahead and throw it out the window. Most people, however, would rather have its stipulations attached.
You wouldn't need to amend the constitution if you actually read the friggin bill of rights. The 9th amendment says that just because a right is not listed doesn't mean it doesn't exist.
You are correct, the Bill of Rights does not include all possible rights. It does, however, currently prohibit the taking of life without due process. In order to change this, you will have to amend the Constitution.
And it's not denying anyone life. It's making their last days on Earth bearable. The states are under no obligation to prosecute everyone for every action that results in way in someone dying. If they were then a lot of people would be on death row right now. No one would work in the defense industry for fear of a malfunction killing military personnel, no one would work in the embedded systems section of the computer industry, heh no one would work for tire manufacturers!
We're not talking accidents here (although, some accidents resulting in death can be prosecuted under law, such as deaths by drunk driving). We're talking about the government allowing the intentional taking of life. Currently, intentional killing is prohibited by the Fifth Amendment. If you want it to be otherwise, you have a lot of work to do to overhawl the protections the Founders set in place. But go ahead.