My parents gave us driving lessons for our 17th birthdays, because we lived on a farm, and there was no public transport. Mostly, they were fed up with ferrying us about everywhere. 
But they did also see it as an essential skill because of where we lived, and had an argument with school about it being a life skill, and useful for CVs and therefore, it should be okay to use free lessons in the 6th form for driving lessons (don't remember the outcome; I'd left by that point.) On the other hand, I have a cousin who didn't learn to drive till her 50s and was planning to move out of London to the back of beyond - you don't need to be able to drive in London, because there is public transport everywhere. So I think a lot depends on where you live as whether it counts as essential or a luxury.
However, while my parents did pay for that (and swimming lessons and other activities like Guides), by the time we got to 6th form, we were expected to buy any non-essentials ourselves, so any piercings and non-school clothes and music and books and cinema outings would have to come from our allowance or Saturday/summer jobs, and we knew that and knew we'd need a good case to ask for things.
We always knew we'd be getting driving lessons - which I wasn't actually that happy about - I didn't take that well to driving, and it took me to my 4th test and being on betablockers before I actually passed, and I'd left home by then, so I failed to save my parents from ferrying me around. I then didn't drive for 7 years, except when they made me, as I couldn't afford to run a car, and didn't particularly want to anyway, and wasn't earning enough to afford it. I did take some refresher lessons when I finally did get a car.
I certainly include "full driving licence" on my CV, and I am expected to drive from time to time in my current role, and when I've been looking for work, it widens the range of jobs I can look at, because I don't have to be reliant on somewhere in walking/cycling distance or on a public transport route.
So, I think it depends where you live as how important it is, but I don't think you are being unreasonable to expect her to make some contribution, because she is going to need to fend for herself at some point, and she'll need to make choices about what she can and can't afford, so she may as well get some practice in now, and if that means getting a job to afford more things, then - that's how life works, mostly.