Speaking as someone who organized folk clubs for many years, I certainly think that a car is a valuable asset to a performer. A performer who has the ability to get themselves to a venue on time is a great help to the organizer who, in all probability, has a day job and (very possibly) family commitments to attend to before getting to the club in the evening.

I had no problems picking guests up at the train station, etc, but it usually meant that they had to get up early next morning so that I could return them to the station before I started work. If the performer was good enough, the fact that they didn't have their own transportation would not have prevented me from booking them on future occasions. But, then again, if the organizer spends too much time ferrying guests to and from the station, performers who have their own "wheels" might just get preference.