Personally, I trust myself not to buy a car that I've heard explodes. You may not, and I respect that. But I see no reason to pay for a service that assures you when I myself require no such assurance.
If NHTSA and DOT operated like UL, products that were less safe probably wouldn't have NHTSA or DOT seals, and those products would be much cheaper. People willing to trade safety for cost would be able to own a car. People willing to pay for safety would pay more. Because of the lower barrier to production, there would be probably much greater choice in car buying, and cars would be universally cheaper. In either case, the choice would rest with individual consumers, who know best their situation and risk tolerance.
First, I can think of very few things that could happen to someone's car that would disadvantage you without disadvantaging them. In the case you mention, the missing tire would render the jalopy inoperable, and so the buyer suffers for their poor choice of vehicle. Any buyer thus factors reliability and safety into their decision, whether or not they are particularly concerned about the safety of others. Plus, now they have to walk, and you probably won't be giving them a ride. A counter-argument to this would be effect on climate, an externality that's maybe best left out for now :-)
Second, individuals may sue other individuals for personal or property damage. So, at least one factor preventing everyone from terrorizing you with their jalopies is that they are liable for damage their jalopy causes to others. The decision to purchase a jalopy includes this risk, disincentivizing the purchase of dangerous (to others) vehicles.
Third, it's already a requirement in most jurisdictions that drivers are insured because of the above liability, and insurance companies could easily deny coverage to unsafe vehicles, as determined by any (private) mutually agreed 3rd NHTSA/DOT-like party.
Finally, because there is no competition in the area of safety certification, it seems highly unlikely that the government's assessments of risk correspond to the risk assessed by the average consumer. The fact that most people choose to drive used vehicles that don't meet the government's latest safety standards feels like implicit proof of this.
My bet is on individuals to act in their own best interest. Likewise, politicians have every incentive to act in ways that benefit themselves and their friends at the expense of everyone else. In both cases I acknowledge the capabilities of these humans to use their own brains to think, but only politicians have an incentive become corrupt.