Look at what is being sold as RTR on-road bashers - simple shaft driven cars like the Tamiya TT-02, Kyosho Fazer and Traxxas 4-Tec. So I would look at something like that rather than a race-derived chassis.
As said above, most public streets are much rougher than you think and have loads of stones and debris on them.
You might even find that an off-road basher makes just as much sense as an on-road one for street fun.