I also agree with broscar. As i posted previously, if you make cars damaged, this would prevent you from reaching the powerups you need to fight as effectively as the other. This means that the first guy who looses a wheel looses the game aswell because he no longer can drive to get powerups. I've seen many arena fights where at the last moment you can flip the situation and become the winner, but with a damaged car it's pretty hard to do so, and it becomes like robotwars or battlebots.
I also would have preferred a system where, no matter what, the cars are invincible and the only points are being given out according to hits you received, but i understand this is a different game from rollcage and it must and should not be a carbon copy.
Still, i can see the appeal that arena mode carries if the destruction is really visible (broken parts lying around, bruns and scratches on the track permanent, etc). So why not finding a middle-way solution?
We plate the cars with destructible elements like shields and parts and bits and so on, and the more you are hit, the more of those parts fall off, but they will not influence the car handling. Perhaps the boosters can get busted and smoke can come up, and wheels can start making sparkles while spinning and the car will change aspect while loosing parts of the hull, but it's only going to be cosmetic. The score will still relay on the point system from hits and there will be car parts lying around, but everyone will have the exact same ability to fight from the beginning to the end of the match.
TL;DR: yes to higher damage level in arena mode, but only cosmetic and winning still boils down to points rather than getting other cars unable to move.