Is it possible to simulate touch without root?
You can potentially simulate a touch with the command input tap x y (e.g. from a terminal emulator, or possibly with Tasker (can't remember if it can do shell commands, but I presume so)), or over ADB with adb shell input tap x y. See https://stackoverflow.com/a/20830258/2141741 (and if you want to connect the device to itself via ADB, https://code.google.com/p/android/issues/detail?id=48126 and associated comments are informative).

This solution doesn't require root (at least not doing it from a computer connected over ADB).