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Based on a game* I think that the root issue is that there are multiple bottlenecks, unavoidable for the drivers, like turning or entering/leaving lanes, forcing them to slow down to avoid crashing. Not a biggie if there are only a few cars, as they'll be distant enough from each other to allow one to slow down a bit without the following needing to do the same; but once the road is close to the carrying capacity, that has a chain effect:
There are solutions for that, such as building some structure to handle those bottlenecks, but they're often spacious and space is precious in a city. Or alternatively you reduce the amount of cars by discouraging people from using them willy-nilly, with a good mass transport system and making cities not so shitty for pedestrians.
*The game in question is OpenTTD. This is easy to test with trains: create some big transport route with multiple trains per rail, then keep adding trains to that route, while watching the time that they take to go from the start to the end. The time will stay roughly constant up to a certain point (the carrying capacity), then each train makes all the others move slower.
You forgot one solution.
Teaching people how to drive safe and smart. Way too many people focus on the car in front of them instead of the traffic ahead. If you watch for brake lights as far up as you can see and let off the gas when appropriate, not only will you be less likely to be in or cause a wreck, you will also save wear and tear on your brakes and use less gas (even more pronounced with regenerative braking).
In addition to the above. When you are driving a route you know well, get the fuck over from which ever side is more likely to be used to turn off. For most highways this means moving left before you near an onramp. Plan ahead and get over before you need to do so you don't have to speed up or slow down to let people in.
That won't help much. By the time anyone notices the roads are slowing down there are six times as many cars on it as it can safely handle. Driving skills will help on backroads, but that isn't where most people are driving. No amount of training can make heavy traffic safe.
This is false and sets a bad example.
It will most definitely help. All it really will take is a certain percentage of people driving smart to make a difference.
As for safety. Heavy heavy traffic at a crawl is much safer than lighter traffic moving at or usually way above the speed limit. Yes, the chances of a rear-end collision are higher but no one is going to flip their car at 10mph. It's the lighter traffic with idiots weaving in and out that makes it even more dangerous and more likely that someone dies when said idiot makes someone swerve out of the way or misjudges and hits something or someone they didn't see coming.
I drive more miles in a couple of days than most people drive all month. I've probably racked up 500k+ miles in the past 25 years of driving. I've been almost run off the road more times than I can count and it wasn't when traffic was at a crawl at the pinch points where traffic merges on to the highway.
Driving safely and smart is essential for other reasons, it does prevent additional bottlenecks (you mentioned one, wreckages), and it reduces the impact of the unavoidable bottlenecks (because the cars won't waste so much time re-accelerating after them). But if my reasoning is correct, most of the time there isn't much that drivers can do against traffic besides "don't use the car".
Laughs in good public transit(rail based is based, but buses are good too), where it can achieve 10~100x the capacity in the same footprint
With rail, as long as you have a good timetable and a robust signaling system, 27tpdph with multiple service patterns is achievable, and >33tpdph if you run just one service pattern, all while having a top speed of 120km/h and an average speed of >50km/h
Railway in general (excluding Line-of-sight based light rail and trams) can move stupendous amounts of people at full speed really quickly due to signaling and mass transit inherently being more efficient in general
Most of the time and places a city doesn't need that capacity. Since your rail cannot get the garbage from my house, or my new bed to the house, we need roads as well. Thus for most a bus running in mixed traffic (remember most roads do not have heavy traffic!) is good enough and a lot cheaper. Where you need capacity a train is really good, but you don't need it.
That said I support trains in a lot more places because trains can run fully automated and thus in the real world can achieve the high frequency people need to choose transit even when a car isn't a problem to own (they can afford it and there is no traffic). This is however just a stop gap since self driving buses don't exist (yet?). In most "first world" countries cost of labor is high and automated trains are thus useful in places where a bus could do the job.