Oh it's much worse than that.

Imagine if this worked, and people could just exempt themselves from the law by carrying a card that says "diplomatic immunity" on it. Everyone would do it, you'd be crazy not to, even if you weren't planning on committing crimes, just to prevent the possibility of a (potentially unjust) arrest or conviction.

And once everyone is above the law, you basically have The Purge, which devolves to more of an Escape From New York kind of vibe, before finally landing in Mad Max territory.

Or, alternatively, we recognize that the card works just as well for the authorities, who can't be charged with anything for ignoring your diplomatic immunity.