Before you go chasing the video driver, there is a cheap test that tells you whether this is a driver problem at all.
Next time the animation stops, do not reboot and do not restart AltTabby. Open the AltTabby config, switch the mode (list vs thumbnail), then switch it straight back.
If the animation comes back, the state that went wrong is inside AltTabby and the driver is probably not your culprit. That toggle is Neil's, from the 0.70 beta thread. A user there had a different symptom, the standard Windows switcher showing up instead of AltTabby, and the mode toggle brought it back without restarting anything. It has not been tested against the animation, so treat it as a test rather than a fix.
While it is in the broken state, open Task Manager, go to the Details tab, and count how many copies of AltTabby64.exe are running. Two copies was the signature in that other report, and it turned up on machines set to launch AltTabby with Windows.
If the toggle does nothing and there is only one process, that pushes it back outside AltTabby and your driver hunch is the better lead.