Two big issues with the death penalty aren’t solved, and may never be solvable.
1 - We cannot know perfectly that a person is guilty in every case.
There is often evidence that exonerates suspects and criminals. Sometimes, we really do know without a doubt, but we don’t necessarily have a process of “we know this person did it because there is no reasonable doubt” vs “we know this person did it with perfect certainty because this person admitted it proudly, ad nauseum, there were cameras, there was plenty of DNA and lots of witnesses. Their own mother testified against them.”
This issue may not ever be perfectly solvable. This means we execute innocent people, too. You can look up famous cases where we executed innocent people. We can’t know exactly how often we do this, but we are aware of doing it regularly.
2 - Our methods of execution are often inhumane and torturous. Some of them char the person being executed. Others paralyze them and put them in a state where their whole body is in excruciating pain but they cannot move or make a sound. There’s a good John Oliver episode on this fact.
We might be able to improve in this area with better methods and technologies, but we’d need federal enforcement to ensure all states are using these.
Also, people who perform executions have no medical expertise, so they wouldn’t necessarily be able to clearly tell we are torturing someone. If an execution fails, we have to revive the victim, who is probably traumatized and tortured, and we try again later.
This process seems inhumane and we definitely would rather get it right the first time, quickly and painlessly, than legally torture people on American soil.