A nation-state level operation that smuggled an exploding lapel pin with high tech shaped charge designed to kill and look like a specific type of gun onto Kirk's shirt without him knowing or feeling the extra weight in coordination with their brainwashed patsy don't make mistakes, certainly not the kind like "uh, did anybody check the type of gun?"

It's one step too far for the theorists to explain why the masterminds wanted you to believe it's a conspiracy, but if it was a conspiracy and not a miracle they obviously planned to make it unbelievable to this target audience.