Is Control Center the only place where one can find the AirPlay button? The AirPlay button that one can use to switch where to send audio/video from an iOS device used to be found to the right of essentially all volume controls (In the Music app, Podcasts app, Safari QuickTime playback, etc). However, I recently noticed that this button is no longer appearing there.  

Perhaps this change was implemented for iOS 7, but I had not really noticed it until very recently. Now it seems that Control Center is the only place with AirPlay controls. Is this the new and only way to access the AirPlay button?
Yes.
It's the only place where the AirPlay Mirroring options can be accessed. Of course, AirPlay can also be accessed from the the player controls while playing a movie.