Every app on my iPhone crashes if wifi is on, but works with cellular

My son's iPhone 5S has iOS 8.1.3 (12B466) and 22GB of free space out of 32GB.

Every single app crashes moments after opening, when wifi is on. Not just 3rd part stuff, safari, settings, app store, stocks, iBooks, etc. If wifi is off, everything works just fine that I can tell. The only exception is the phone app, the phone app works all the time and I am guessing it has something to do with the phone app not running in the same space as every other app on the device, right? It's part of the reason that iPhone don't allow tighter integration between 3rd party apps and the phone app or something like that.

I can get about 2 seconds inside the settings app to switch wifi on or off, but no time to get to anything else when wifi is on.

I have temporarily disabled all the restrictions on my son's phone (normally there are controls only sign-in/out of accounts, no access to safari, app store) and of course I have reset the phone and powered on/off multiple times each.

I have not done a full reset to either reload from scratch or from a backup; that's what I am trying to avoid.

Oh, and the phone was new in December; everything was loaded clean at that time, not from a backup. (The phone has been restored from backup a couple of times in the past 2 months. [Access to your phone is a right given by your parents, and noncompliance results in an "erase device" command sent from iCloud.com, followed by a restore once things are brought back in order!]