This machine came to me with Windows 10 on it, which had no issues with the touchpad, suggesting that whatever is going on isn't hardware related. On qubes, however, I've used xev to identify that the touchpad doesn't send pointer movement events (generally speaking), unless I also hold a finger over either the left or right touchpad mouse buttons. If I do place a finger on either button, no matter how light or hard I press, the pointer will move as I like. It will also operate as desired if I place a finger somewhere else on the touchpad, but less reliably than on the buttons. Tap to click (left and right), and multifinger scrolling work perfectly no matter what.
Holding the left/right mouse buttons of my external usb device doesn't influence whether it works or not, so perhaps the button press isn't as important as having a larger surface area to complete an electrical circuit? I've also tried using the synaptics driver, but it actually seems less functional than the libinput default.
The only exception to the rule is that, rarely, the pointer will catch a movement request and jump before immediately reverting to being nonresponsive. This doesn't seem entirely dependent on "sweet spots" as I originally though, but I still think there is some correlation between it deciding to work and where I try from on the touchpad.
All help is appreciated, and if I can provide more information, please just let me know.
Much thanks,
Adam