If you ever see anything whatsoever that can be improved about how things work on Apple platforms, it's always good to file an enhancement request through Apple's bug reporter site: http://radar.apple.com or http://bugreport.apple.com (both links end up at the same place).

And if that's a bit too techie for you with all the form fields that are designed for developers, there's a consumer-friendly site that similarly allows you to submit suggestions including enhancement requests at https://www.apple.com/feedback/ .

Completely agree. I've been an iOS developer for about five years now, and they really do respond to radar reports. One of the better companies on that front.

IIRC, there's a setting for it on the phone, somewhere in the Messages settings.

All I see is the option to forward text messages to selected devices (which would either always or never show them). Thanks for the answer, though!

Right you are... I was thinking of the "Repeat alerts" message notification setting.