I actually did a little experiment since I experienced this problem also. What happens is that the car you made would run on level blocks and not on terrains.

Now what I did was that I made a nearly flat terrain (still with few bumps here and there) and a wide level block floor. After that, I placed a bridge between those two. I obviously placed the car on the level block and assigned the template actions to the car and the wheels. Voila! I managed to drive it around the level block, on the bridge and over the terrain. The car did manage to run smoothly over the terrain but then it gut stuck eventually.

Based on this experiment, the template car object CAN run on a terrain given the conditions I mentioned above. Perhaps we can start figuring things out from here.