Thanks for the further reply. I now see what you mean about where you placed it.
I have tried what you suggested (assuming I've done it right) and still no joy. It's starting to get silly.
Step one: Locally, inside the "web.config" file I added an exact duplicate of your example code inside the "handlers" of "system.webServer". I then uploaded this to the "live" server, replacing the previous "web.config" file. In case I needed to restart the Sitefinity system, I did that thing where I made a minor, non-consequential change to the "web.config" file and saved the change... But I don't think the system restarted, as browsing pages continued as normal without a pause.
Step Two: On the "live" server I copied that new redirect code and pasted into the same location within "Extensionweb.config". Again, I did a minor, non-consequential change too and saved it. Then browsing the site's front-end there was an obvious pause as the browser was waiting on the server - presumably doing the system restart.
But, when applying "/Telerik.Web.UI.DialogHandler.aspx?checkHandler=true" onto the end of my domain's URL I was once again greeted with "HandlerCheckOK" as opposed to the "not allowed" web page I was expecting (and yes, I had pre-created a new Sitefinity page that used the same URL as your example).
So I'm somewhat bamboozled.