iPhone 5S broken screen root access without iTunes authorization
Software

iExplorer claims to grant you root access to iDevices without requiring a jailbreak:


  With iExplorer, you can access more files and folders without modifying (jailbreaking) your device than ever before. Need to access photos on your iPhone or iPod? Can do. Want to access the files and folders of your device's apps? Done. How about browsing through your iTunes backups? Check. What about the media folders containing your iTunes purchases? We've got that too. What about files in apps that are synced to iCloud? Yes, even that.*


iBrowse is a simillar, free software by the same company, but it doesn't grant root access (which I'm not sure you need in order to access photos).

The "Windows" Method

From an answer on Apple's support forums:


  I plugged the phone into my Windows Computer and when it recognizes the phone is plugged in, a dialogue box pops up that asks if you want to import pictures and videos. You click import and it will pull everything off the phone and place it in a folder with the date as the name and put it in your pictures or my pictures folder. From there you can access the photos &amp; videos!!


I'm not sure this works either, update in the comments so I can modify my answer.

Old Guide

I also found a guide, written in 2008, and doesn't involve any 3rd party software: (Not sure it still works)


  
  Connect your iPhone/iPod touch and launch Applications >Image Capture.
  Go to Preferences >General.
  Under Camera, select Image Capture in the “When a camera is connected, open:” option, and then click OK. Now when you connect your
  iPhone/iPod touch, Image Capture will be launched and you can choose
  where you want to download your images.