Ok, I’ll put it differently.
Imagine you are a dayan in a capital case. There are two kosher eidim, and everything has been met. But, a bas kol says, “he didn’t do it.” Would you execute the innocent “murderer”? What would G-d want you to do?
When a goyish court uses standards which are allowed by G-d for them, the same is true. G-d wants them to execute the people they find guilty under standards which are acceptable for them. I am happy when G-d’s will is done. I hope you will be happy with me.