what specifically would a GPS change about the planning and execution of a flight to remove these errors
I don't think anybody mentioned "mandatory", but a GPS gives you constant guidance along a track.
It also provides a gross error check because you load your route into it and the route is displayed overlaying the appropriate bit of the map. It's possible to mis-program but very hard to do so and not spot the mistake at this stage.
Anyway, this is a QXC which under present regs is navigated with dead reckoning, so this isn't relevant as things stand. I think it should be brought into the PPL because human intellect will never get better, the training (in today's ATPL hour builder "just passin' through" climate) will never get better; so we will always get these big busts, and on the assumption (which may or may not be reasonable) that the powers to be would like to improve matters, something else has to be done. People will always jump on anybody suggesting this but TBH I don't see anybody having a better idea.