I agree this policy feels kind of... excessive?
In practice, it's been imperfect. Gate agents have been making numerous announcements prior to and during boarding, referring folks to the ticket desk for individual boarding passes as needed. Southwest has also had staff checking individually with the parents who line up early in the family boarding line. I only recall seeing one group turned away at the jet bridge, but can't speak to how often it actually happens. Because I travel so frequently, I tend to be among the first to board, missing any later boarding drama. Also, my weekly travel is limited to three or four midwestern airports, and I have little experience with what's happening in other regions.
When I travel with kids, I generally stop by the self-help kiosk on the way to TSA and print out all boarding passes, anyway. It's a good back-up in case my phone glitches, and the kids love carrying their own passes. And, they love using the boarding passes to navigate us to our gate!
I will say that airport staff are stretched thin. If I can do something small to make their day smoother (which, in turn, makes my travel smoother), I'll do it.
Back to my first point, though. As a policy, it feels excessive.