No
Currently any site that needs to be moved to stable websites section from pending or new sites section has to ask any admin or moderator to do it.
So its equal work. Infact it will reduce their work load as they will have one less section to clear from SPAM posts.
Ok what you are suggesting is something like this.
Of course, only admins can move Topic from New to Stable section.
And Stable section should have "Reply Only" permission. So members can discuss about already approved stable site but themselves can't make new stable sites 
This is what you were trying to propose?