Why were they even introduced?
I thinks it is debatable whether they are a horribly bad idea.
They permit the OP to moderate the discussion and filter out the trolls that do not contribute to the topic without wasting the moderators' time to do this (and sometimes the OP might not be satisfied with the fact that not all trolling posts will get deleted). If I'm not mistaken Moderators/Administrators can still see the deleted posts.
On the other hand it makes some scammers' scamming easier so I think the best solution would be to permit only the very-established users (similar to the way the Ignore button works) to start self-moderated topics OR maybe introduce a small button that when clicked, would let others see all the posts that have been deleted in said topic.
The benefit from it is very small (some guy gets to feel important) and the damage serious (all the scammers get to control the flow of conversation in their scamthreads which is all they ever want and all they ever need to succeed).
It's impractical to fix this, and none of your palliative proposals work (there's no way to establish who should get this ability; if it works anything like the ignores then it's as dysfunctional as it gets; there's no code changes ever going to be implemented simply because it'd open the Pandora's box of the ~half a million dollars' worth of "forum improvement funds" for a forum that still runs on borrowed server space etc).
So debatable it may be, as anything can be debated if people get bored enough. It's also a fact.