You're completely ignoring the fact that consumer choices about automobile safety have externalities beyond the buyer. Sure, maybe you and I care enough about auto safety standards to buy a vehicle that isn't dangerous, but I don't trust other consumers to necessarily do the same thing. I don't want to be sharing the road with some corner-cutting jalopy that's likely to throw a tire through my windshield while we're passing each other on the highway, and I definitely don't want a market that permits such low-quality products to be operated in public spaces.
I can think of a few systems in place that mitigate cases like the one you mention.
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.
None of your defenses work outside of a thought experiment where every actor is rational. Real human beings are anything but, and there's a long history of precedents for why all of your positions here are bad ideas (and why regulatory bodies like NHTSA, DOT, and other regulators became necessary).
And your arguments rest on the assumption that politicians and voters are rational, surely?
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.
