I have saved for Web in PNG 24 and checked the transparency box - it does normally work with transparent background, but with a transparent graded shape it doesn't work. What's wrong there?
Change your preview mode to Original
Right now your image render is previewing in GIF format. If you left all of your settings the same right now and just hit save your file will be fine. It is just the preview that isn't showing your transparent gradient. The actual file will have the transparency.
The optimized preview shows GIF because the first selection for output is on GIF (or maybe it uses your last option, I am not sure). You can go back to the Optimized tab once you have selected Original.