Okay, so I'm not an expert, but hopefully a B.S. in nuclear engineering is enough qualification to answer some of the questions posed.

1st point; obviously the fantastic thing about nuclear power is the high energy density of the fuel, that's what makes it so attractive in the first place. However, a reactor sized for cars would definitely need some sort of refueling, at that size you'd already be very hard pressed to reach criticality with geometry and mass constraints. Something so small would be impacted quite a bit from burnup and fuel poisoning.

2nd point; the name of the game for nuclear is to contain radiological releases. While we accept other forms of pollution quite readily, there isn't really any amount of radiation discharge that the public is willing accept. Call it "too dumb" if you want, but the reality is that any risk of nuclear contamination is poorly received, just look at the public backlash pretty much any commercial reactor faces. This is despite the fact that coal-fired power plants release more nuclear contamination than nuclear plants. Millions of vehicles susceptible to high-impact collisions? That's a non-starter. There'd be no way to guarantee that radiation wouldn't be released to the environment, and I'd go as far as to say it'd be inevitable. Like you said, this wouldn't be another chernobyl, but we'd still be dealing with ridiculous amounts of activity. Your guess of it being less dangerous and local than current crashes is completely wrong. Any involved in the crash? Dosed strong enough to get acute radiation poisoning in minutes. First responders? Same deal. And then the winds/rain would scatter much of that contamination to the surrounding environment, where it would linger at dangerous levels for thousands of years. We're not even talking about the long-term problem of iodine or cesium, exposed nuclear fuel would give enough of an alpha dose to kill anyone near it.

3rd point; the waste solution is so tricky I don't even want to get into it. For all intents and purposes however, right now only one country in the world has a long-term waste storage facility, and it's Finland. Even if experts could agree that deep-geological storage is the best option, such an option is politically impossible for most countries. On-site storage is far from "trivial," and the dry cask storage employed by the US requires a pretty significant amount of engineering and monitoring. As for reusing fuel, it's not as easy as throwing spent fuel in a new reactor and calling it a day. It requires a pretty significant amount of politically contentious infrastructure and reprocessing, of which only a few countries (France, Japan I believe) have.

I'm a huge proponent of nuclear power, but if I'm being completely honest, making the case for it is getting harder and harder, even when only considering the technological aspect. There's lots of new and exciting designs, but I wouldn't bet on them to make a huge impact in the global energy makeup moving forward. Even for large scale energy generation, nuclear is heavily burdened by cost and political opposition. The idea of fission technology being usable as a fuel source for personal transport vehicles is laughable at most.