Nope, cuz in every movie, he starts off with one in hand.
And, seeing that, according to the Skill Balance Sheet posted up above, without assigning STUNT DRIVER a skill point value, he currently has a point value of 11, if STUNT DRIVER were to be given a value of 5 (for example), he would only be at a total of Balance Score of 16, which puts him in the "comfortably strong but not completely overbalanced" area with Doug and Dave. I personally think that if STUNT DRIVER were to be given a Balance Score, it would be a solid 4 in Orange Range, as it's technically giving Mel the ability to use a FREE ACTION (at a score of 4 in the Orange Range, where STUNT DRIVER is currently assigned) to move in and out of cars without penalty. If this is the case, then Mel would be given a Balance Score of 15, putting him company with Josh, Wanda, Ned, and El Cholo.
I do see what you mean, however. He basically gets to start with 2 Dice @ 3+ To-Hit to kill Zombies. In my gameplay testing, while he did come out on top of nearly all situations at the beginning, it didn't feel all too broken as nearly all of the scenarios I played him in were without cars (a car would basically make him crazy OP in terms of XP gains, so I wanted to see how he'd do without it). It basically felt like I was guaranteed at least 1 Walker kill per turn.
Maybe I'll give him Destiny instead as his Blue skill, instead. Granted, it'll give him a score of 18, he won't start with a weapon, putting him in company with most of the higher end custom characters (which was what I was trying to avoid). I'll test it out tonight and let you know how it's going.