Excellent video. So easy to go down that road to disaster. And not faulting the pilot.
But in retrospect, one available option seems lacking in training and practice. If confronted with the situation where both ATC and onboard avionics give a terrain/low altitude alert:
1) Full power and pitch for Vx.
2) Use onboard terrain depiction - if available - and choose a heading toward the lowest of the indicated high terrain, possibly including an immediate 180° turn.
3) Declare an emergency and sort things out with the controller.
In this particular case, such an action would have put him in VMC conditions almost immediately. I know that’s easier said than done, due to the gradual onset of uncertainty of the plane’s proximity to the higher terrain and fixation on the GPS programming and intermittent failure to depict the route. But training should include just such scenario-based situations so the above actions become almost automatic.
edited to add: Such an action would be highly unlikely to lead to any enforcement action. Maybe just a call to the facility to explain what happened, and a NASA report filed just in case. The very last things a pilot in this situation should be worrying about.