I think it bears mentioning that these are pilot programs and being tested in a couple of jurisdictions. It's not standard practice yet. What it will probably develop into is more of a system whereby officers would have an NFC reader in their phone or device and you would transfer your drivers license and/or insurance information over to the officers NFC-capable device, and he would have limited access to view the information he needs without storing the data on his device permanently. Of course, in order for this to work, Apple would have to open up their NFC reader to developers, instead of locking it to everyone but Apple Pay.
I have used the Progressive app on my phone as proof of insurance with officers. This IS actually standard practice in many states (Tennessee, for example, completely allows digital proof of insurance). Every time I have showed proof of insurance to an officer this way, they have never taken my phone back to their car. They saw that I had insurance and that was good enough for them and they didn't ask any other questions.
As for the "battery is dead" issue, that should be a non-issue in the car if you have a car charger for your phone.