|"Often the only way to get them to sit up and take notice, after a couple of friendly initial emails, is to make them aware that you can, and will, contact their hosting provider and request them to shut the site down if they do not comply with your request."|
The legal basis for that is slim to none, and slim skipped town years ago. This is not a DMCA request.
But assume you are successful with a threatening demand and the web host shuts down the directory.
This is what happens next:
The directory owner sues you for tortiously interfering with its web hosting contract. Not only do you have liability for any damages, but also punitive damages.
In other words, this is not a well thought out strategy.