I'll use an existing answer of mine, as the advice is almost the same. But one is a friend, the other one a colleague. Makes a big difference.
Walking on eggs, because you still need the ride, and can't annoy this "superior", I'd use the fact that this person seems always on a hurry. And ask them if I can drive, so that they have more time to use their phone, ou do whatever they need, and can't (or shouldn't) do while driving. You offer them a service, they can say yes/no but don't feel offended most of the time.
Show them the advantage that they have letting you drive. But never point out what they do, they'll understand it as "you're a bad driver" anyway, no matter how you say it.
If needed, a small white lie can be used. Explain that you would be pleased to drive because it's something you want to do to thank them for the ride (do your share of the ride), and that you haven't done that for a while, enjoy it, and need practice. It's almost like asking them for a favor.
I always drive when with someone, and they enjoy the free time they get because of that.
If you drive safely, you lead by example. What you can't tell, you show.