I was standing behind someone in a queue for boarding a flight whose phone died from lack of charge as they were about to hand it over to be scanned - so having your boarding pass on an e-ink display might actually be a pretty good idea...
Edit: downvotes don't make that any less true. The situation the OP was observing could very well have been that the person had checked-in via phone and when boarding was starting, the phone's battery died.
Because she doesn't necessarily was even near the airport when she checked in. Nowadays you can check-in online or via your phone up to 24 hours before your flight starts.
And you never see any paper. Yeah, that sucks when your phone runs out of battery. In our trains, the conductor has now USB chargers in case such things happen.
Technology fails, we fail. It happens. We manage to adapt.