There are many other possibilities, please stop projecting your close minded views onto others.
How do we get it so murderers aren't a fiscal burden to the state? The only way I'm thinking of is slave labor, but if you have them do construction work, this runs into 2 problems:
1) They would be taking jobs away from working class Americans.
2) They are going to do a worse job than those working class Americans.
Where for example does your consideration for wrongful convictions come into play? How many innocent lives is an acceptable cost to ensure all murderers are killed by the state?
Most people don't really care about the lives of strangers, this is why they aren't willing to spend $1/day to save a kid in Africa via sponsorship. If people aren't willing to sacrifice $1/day to save an innocent child in Africa, I don't know why they would be willing to sacrifice $20/day through collective tax money to save some adult that probably committed murder.
I don't know the proportion of innocent deaths that is acceptable to execute all murderers. But I have a high tolterance for risk, so that number is pretty high, especially since I don't want to spend money to save lives that are obviously innocent (like children in Africa). If I'm unwilling to save kids in Africa, why would I be willing to save people in the US that probably committed murder?