Question: Sometimes I'm on call for work overnight and would like to know a quick way that I can silence all notifications except for the call ringer. The closest thing I see is allowing calls from everyone on do not disturb but that will allow messages to ring as well.

Answer: 
Actually, Do Not Disturb is indeed your solution as it blocks ALL notifications EXCEPT the calls which you specifically allow. It will block message notifications. 

After you have configured the settings in Settings -> Do Not Disturb, all you need to do to enable it is swipe up from the bottom of the screen and tap the moon icon. 

You can configure it to only allow certain contact groups, favorites, all callers, and/or repeated calls.
You can also head to Settings->Sounds->Text Tone and set tone and vibration there. You can configure texts for no sound and/or no vibration. Similarly, you can allow ALL calls (from any number) to ring as you have set in Settings->Sounds->Ringtone

I was just dealing with the same issue as I expect and need to hear work calls (any number) but can't have my phone buzzing from a bunch of texts all the time.