I need to know why. SF used to provide a key-based championship mode, and it's not in Arcade Edition anymore. I was told that it let cheat happen, but that could be solved without cutting this feature off.
The Championship Mode DLC you are talking about is tagged as Downloadable Content (DLC).
Super Street Fighter IV: Arcade Edition (DLC) is also tagged as DLC.
Capcom decided that they wanted two different experiences, so they released them as separate DLCs. Meaning that the features from the Championship edition are not available in the Arcade Edition.
In consoles, there was the Super Tournament Mode DLC for Super Street Fighter IV, which enabled a "Tournament" option, that is similar to the "Championship" mode that was also released as DLC for Street Fighter IV. Looking at the screenshots for Super Street Fighter 4: Arcade edition on Steam, I can see the "Tournament" option there, but it looks like an XBox 360 screenshot.