Can you, what I'm calling, "snapshoot" videos of activity on one's iPhone screen?
With iOS 11, you can use Screen Recording built in to the OS.


Go to Settings → Control Center and add Screen Recording.
When you're ready to record, swipe from the bottom of the screen to show Control Center, then tap the Screen Recording button. 3D Touch the icon for additional options such as recording the microphone.
Continue using your device and your display will be recorded.
To stop the recording, tap the Screen Recording button once more. The video is saved to your camera roll.


If your device is jailbroken, you can use Display Recorder which performs similar functionality for older versions of iOS.