Another advantage is that the app can be updated by the server when updates happen using a Push synchronization. So an app with few feeds would very rarely wake up the device to sync things in the background.
That said, the first versions of my RSS reader app Feeder did just this with a server and I later moved to check all feeds locally on the device due to the massive maintenance burden it actually was.
The code for the server is still available in Feeder’s repo if you dig in the history to this place:
Though it’s probably not very useful. The API was built for a minimum communication between app and server, so it doesn’t serve up a combined RSS feed.