Goofs

Warning: Spoilers Police always check for gunpowder traces on hands to verify who is the shooter. It would have shown who is guilty and who is innocent.

From an active police detective: The police (specifically the medical examiner) would likely only test for gunshot residue if they saw stippling, powder burns, or defensive wounds, which helps establish if the shooter was standing close by. But they wouldn't check this with suicide in mind. You get the residue by being close to the gun when it goes off, and it doesn't matter which end of the barrel you're on. So the presence of residue on her hands wouldn't mean she had to have fired the gun. There is no plot hole here.