Do mods actually have the power to deprive the user of the usual notification by PM?
Yes.
ON another SMF forum there is a post:

As a global moderator I click delete and get this dialogue box:

I click ok, and the message is gone.
No advice to the user.
Which is why I know the moderators are making bold face lies about what they can and can't do here. It only takes minutes to set up an SMF forum on a website (and it's free!) - give it a go and see what you can and can't do Vs what you are told can and can't be done on an SMF based forum.