Ok, and you don't think that if they weren't explicitly concerned with the appeals process or the questionable mental health of the prisoner, that they wouldn't just execute them immediately?

You are hopelessly naive.

They're not concerned with whether it hurts people or there simply wouldn't be death sentences at all.

What they're concerned with is convincing heavily propagandised naive dumbasses that they're acting with good intentions, and implement the bare minimum that they're politically pressured into implementing in order to achieve that without fundamentally changing the core of the system itself.