TMZ is a brand of TimeWarner, so at least any rights fee would still be under the AT&T umbrella.
This channel has been an afterthought for a long time. It really lost its direction when they added long form talk shows to what was originally CNN Headline News.
Are you sure they still don't get a carriage fee? I think they, with the larger cable companies, sell the CNN and other Turner Channels in a complete package. I know HLN, when it actually was Headline News, generated additional revenue through OTA syndication, but that dried up when they changed formats. I think they could record the 30 minute format and run it for four hours, at least in late evening/overnight and if something breaking happened have the main CNN team cut in a segment to cover it. It was wonderful to be able to tune in at any time and get the full range of news in 30 minutes. Robin Meade proved her worth, but for the rest of the day a return to the old format, even if it aired taped for significant portions would seem to be worth a try.
HLN is the only channel that doesn't get a carriage fee.