Is it possible to read SMS with broken touch and disabled usb debug?
If your device has a USB OTG port (My Xperia Z1 Compact and Nexus 7 do, and I believe most recent devices do) you may be in luck - get a USB OTG adapter (mine was from a pound shop - if your in the uk try Poundland) and a USB mouse - plug them in to your phone and you can then control it. 

Its a bit of a faff at first but better than nothing.