Tried at home with an official Apple Lightning cable, plugged the iPhone into USB and my error messages were this:
9/21/16 11:16:55.689 PM usbmuxd[83]: AMDGetPairingRecordDirectoryPath (thread 0x7fff74c7f000): Could not chmod /var/db/lockdown/: 2 No such file or directory
9/21/16 11:16:55.690 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:55.690 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:55.736 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:55.736 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:55.736 PM usbmuxd[83]: _SendBUIDToDevice (thread 0x700000094000): AMDCopySystemBonjourUniqueID() failed
9/21/16 11:16:55.754 PM iTunesHelper[315]: Entered:_AMMuxedVersion2DeviceConnected, mux-device:1
9/21/16 11:16:55.757 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:55.757 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:55.761 PM iTunesHelper[315]: tid:7713 - unable to query device capabilities
9/21/16 11:16:55.797 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:55.797 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:56.217 PM iTunes[1301]: Entered:_AMMuxedVersion2DeviceConnected, mux-device:1
9/21/16 11:16:56.219 PM iTunes[1301]: tid:1a03 - unable to query device capabilities
9/21/16 11:16:56.426 PM iTunes[1301]: ApplePushService: APSConnection being used without a delegate queue
9/21/16 11:16:56.651 PM PTPCamera[1300]: --> HandleMessageFromICDD
9/21/16 11:16:56.726 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:56.726 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:56.827 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:56.827 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:56.828 PM iTunes[1301]: failed to connect to USB device 0x7fbe7a6770f0, UDID [REMOVED FOR PRIVACY] (0xE8000003)
9/21/16 11:16:57.403 PM com.apple.usbmuxd[83]: WriteDataToPath unable to create file /var/db/lockdown//SystemConfiguration.plist.tmp: No such file or directory.
9/21/16 11:16:57.403 PM com.apple.usbmuxd[83]: CreateAndStoreBUID WriteDataToPath failed: No such file or directory
9/21/16 11:16:57.404 PM apple.DevicePropertyReader[1302]: AMDeviceExtendedPairWithOptions (thread 0x70000019a000): Could not create system BUID
9/21/16 11:16:57.406 PM apple.DevicePropertyReader[1302]: Pairing device failed for [REMOVED FOR PRIVACY]: 0xe8000003 (Could not allocate a resource.)
Later on:
9/21/16 11:25:26.326 PM sharingd[271]: 23:25:26.325 : SDActivityAdvertiser::continuity:didFailToStartAdvertisingOfType:withError: TetheringTargetPresence (The operation couldn’t be completed. (com.apple.identityservices.error error 200.))
9/21/16 11:25:26.326 PM sharingd[271]: 23:25:26.325 : Tethering: Target failed to start advertising IDSContinuityTypeTetheringTargetPresence (Error Domain=com.apple.identityservices.error Code=200 "(null)" UserInfo={NSUnderlyingError=0x7fe1fb022e80 {Error Domain=NSMachErrorDomain Code=1 "(os/kern) invalid address" UserInfo={NSLocalizedDescription=Unknown error.}}})
There's a lot more usb-type errors.
On the iPhone I got the "Trust/Don't Trust screen"
On the Mac I got:

Linc is great.
Well, that's different. Going to try to reboot again.