Is there a music app that lets me start/pause playback by tapping anywhere on screen? I'm looking for an iOS app where, once the music is playing, I can touch anywhere on the screen to pause it, and then start it again.
The CarTunes Music Player app does what you need - there are no buttons, just taps and swipes for all music controls. 

Tapping is play/ pause, swiping is next/ previous track etc. Even volume can be adjusted by holding and dragging up/ down. 

The app is not free at the moment but does fluctuate in price and is sometimes even free, as can be seen in the App Activity sidebar on App Shopper