Oh, maybe I'm missing something? As far as I know, the only consequence is that the temperature rises, and you can't use the turbo until it cools down again. I've never experienced having to stop at the side of the road, despite the many hours I've spent playing the original game. Do you know of a video where what you describe happens? Maybe I can implement it, based on that video.
In Original mode, however, high temperatures do have a consequence: engine damage and the risk of blowing it up prematurely.