We can certainly agree that if an innocent person is executed that is just state-sponsored murder, right?

The jury system of the US is a good system but it’s certainly not immune to mob mentality. It’s certainly not immune to appeals to emotion. It’s certainly not immune to any number of the things that make humans human. This is partly why we have an appeals process. If we are going to put someone to death, I think it’s better for them to be given every single avenue possible to prove their innocence so that we aren’t murdering someone innocent.