>Hold a monthly lottery of the population, ...
The point you're making is that living in society each day is a ticket in lottery where we could be falsely accused of a crime, falsely convicted, and unjustly punished. That's horrific all around, a death penalty is a factor but not even the biggest. On the contrary, if we need a death penality to make someone like your audience care about that horror of punishing innocent people then perhaps there is a beneficial purpose? Maybe executions make people care about bad things done in their name in the name of justice?
>Police dropped evidence into your pocket? Try them for attempted murder.
Yes, naturally. Some archaic pockets of government do this, sort of! What comes to mind is a specific public officer in my state who is personally responsible by law for breaches of privacy in her office. Consequences like that focus the mind! So much so, in fact, that those consequences come to dominate the public officer's thinking. I don't disagree, but forget 'defund the police'-- you're going to need a lot more police to enforce the modern criminal code at the current level of enforcement if police are threatened with death for misplacing their notes , memories and paperwork.
>At your conviction lottery is held of the population and winner is your executioner.
Again, the result here would be that some people would be assigned an executioner who would show them mercy. This too has analogues in our modern systems. In some parts of the world the victim's family can stop an execution if they feel the victim has been adequately avenged-- not a bad result! In the US context governors-- arguably the people responsible for these killings-- can and frequently do commute sentences out of compassion or lack of clarity about the crime.