Maybe you could show your newer hammock/tarp and relate how a camping mattress is possibly supplanted by that system? You could discuss temperate use?
Even at temps in the high 60˚F range, you will get "cold butt syndrome" if you sleep in a hammock without insulation. I use down underquilts in various temperature ratings as bottom insulation in my hammocks. Some folks do use sleeping pads (camping mattresses), though.
My newest hammock, the Warbonnet Blackbird XLC, was purchased with a double bottom layer option, making it easier to use a sleeping pad as bottom insulation. A sleeping pad certainly can be used in a single-layer hammock, but it may be difficult to keep in place while sleeping.
The benefit of using an under quilt, as I prefer, is that it is hung under the hammock and stays at full loft... your body weight is not compressing it and compromising its insulating qualities. With an under quilt hanging snugly below you and an over quilt on top of you like a blanket, you are basically inside a cocoon of down insulation. It's as comfy as it sounds. And, also important, ultralight in weight and extremely compressible for efficient packing.
I don't have links to any photos of my Blackbird XLC hammock or Superfly tarp, but you can see them on the Warbonnet Outdoors web pages. (I wish we could upload photos for posting on these forums...)