I found a guide on reddit (Apparently I'm not allowed to post links, but it'll be the first result if you google "How I managed to use my Android device with a broken touch screen")
It's not exactly simple, but it should work.
Unfortunately, I discovered that in order to use ADB shell commands, a pop-up window should appear on the phone to allow/disallow USB debugging. (USB debugging is enabled already, but there appears to be an additional 'authentication' step)
Unfortunately, I can't use the phone's touchscreen, so I can't unlock it or tap the notification that will allow me to authenticate the phone.
Is there a way I can get around that authentication step?
The device is unlocked and rooted.