I mean this in a very kind way, but you've only been with the man a year, only met the kids 6 months ago, haven't experienced living with him yet and yet you seem to think you are strongly entitled to decide how your partner should act in relation to his ex.
Frankly, start this way and you are opening the gates for years of misery. You are totally overstepping your position. It is not for you to conclude that if you give in to her demands, it will mean she can use the excuse of stopping contact any time. You have no idea if that is true or not.
You are implying that this is likely to happen but yet so far, it seems that contact has been regular. You have also hinted that one child will be moving with you, which is quite odd considering your situation.
My gut feeling is that one of the child made a mention about your driving to their mum that's what triggered the whole thing. Probably the reason why she told the kids, not in a 'I forbid you to ever go in her car' but in a 'mmm, maybe it might be safer if you just hop in daddy's car'.You might think you are a wonderful driver but supposedly we all think that!
The bottom line is that you are absolutely right that she shouldn't dictate who drives the children when they are with their dad, but when it comes to being worried, mums don't care any longer about being fair to the other partner. There is a very easy alternative and that is that indeed, your partner takes them all in his car, but clearly, your position is about principle rather than practicality.
If you'd been together for 5 years, and this was the nth time she was making threats of cutting contact, then maybe I could see your point, but as it is, you're whizzing into the life of her kids at high speed, and the break is still fresh so starting of by confronting her over something that most likely will blow over in a few months if starting ww3 which will pollute your life much worse than by driving alone when you go out.
Back of and focus on building your life with your new partner and relationship with his kids.