I can so sympathise with the ex as I found myself in the same position than her and it is horrible. My ex got his driving licence late, drove very rarely so gained very little experience and yet drove like a complete over-confident idiot. I used to be feel sick with worry when he picked them up, and then sped up like a maniac, probably making a point.
He too had suffered from depression and threatened to end his life a few times. When I read on the news shortly after that that a father and two children had been found dead in a car after their dad had asphyxiated them with him during a visit, I started having serious anxiety attacks when they were with him.
Then he too had an accident, and tried to minimise it, insisting over and over that it wasn't his fault even though the evidence indicated differently. So yes, at this stage, I told him I couldn't continue with the sickness I felt every time they were with him and said I would rather he didn't drive them. He went berserk, but thankfully for me, his -very old- car broke down and he didn't have the money to replace it. I never felt so relieved in my life.
They are now teenagers and have very few reasons to be in his car as they never do anything, but I still get nervous when they do and feel grateful when they come back safe. I remind myself that it is not too far away when they will be driving themselves and I won't be able to do anything about it.
In the end, if she is feeling as anxious about it as I was, she could decide not to late them go at all if he refused to abide to her request. He would then need to go to court and she could very well have a case. I totally understand your frustration though because ultimately, it is not you she is worried about. I personally would hurry to learn to drive.