I don't know if that makes a significant difference for that purpose.

It's not like places where weddings take place are conspicuous enough that you couldn't tell the difference between them and a normal govt agency, and it's not like people don't know it is a very bad idea to sign documents where they do not even understand a single word.

The paper trail of a vanilla (let alone international) marriage is already big enough that I doubt criminals would want that, unless they have corrupt officials assisting them, which is a problem these measures do not address.