Why not have a push notification to connected phones when speaker starts playing from voice control?

The Sonos app is what makes the actual connection to the system, so it’s rather a chicken and egg issue. 

What phone is it? You could always leave the Sonos app running in the background. Depending on the operating system it may already have the facility to display the current room/source when you take it out of your pocket.

Minor detail: In a system where multiple listeners/users are active, which player should post to your phone?

And should it ping every phone on the network? 

Push notifications are slow, and unreliable. It was one of the reasons I pulled my old Alexa integration, as the 7 second Alexa timeout was often too short when I had to do a Push, a local network scan, Zone determination, and multiple-device detection. The OP’s scenario would have to do similar work, and somehow hold off the completion of the Alexa command (and avoid the timeout) until the user responded to the Push.