Now if you want to convince me otherwise.... then give a detailed explanation of how you see the transition occurring. Pay particular attention to human biology. Hint: The Air Force has a *ton* of the data on this.
I take the challenge not to convince you. but as an opportunity to think about the subject. You said you are correct so many times I start to believe it myself :-) Also, I think a lot of progress can happen, so the focus is on where it might get traction.
Where? It might be a place like Nevada, or the USA in general. Pros: long stretches of highways; very limited permissible top speed (usually 55-65mph, very rarely up to 80mph); mostly disciplined population with respect to speeding, tailgating, driving speed variations, aggression, DUI etc.; good quality wide roads, generally OK weather conditions and generally young and reliable cars. Cons: lawyers.
Type of road: your suggestion (HOV lanes) seem to imply mostly suburban traffic. While I agree with the HOV lane approach, and indeed it has economic benefits (much smaller distances among cars mean higher throughput, therefore higher speed at the same or lower fuel costs). But the suburban setting may not be the first area: there is rush hour traffic, bumper to bumper traffic and urban settings (otherwise there wouldn't be a HOV lane). Google started autonomous driving on the highways, and I think that's the most likely type of road where it all starts. It's boring and it gets dark (unlike in the city where most driving is during the day, and/or roads are well-lit). Drivers are prone to fall asleep. It's for a reason that highways are worth the investment of roughening the road sides to wake up the driver. Highways are rewarding from a technical standpoint too: the traffic patterns are much more predictable and much less ad-hoc, and if it IS ad-hoc (e.g. accident ahead of you) a millisecond-level steering or braking action may just save lives. So much so that the technology is being built continuously: sensors to keep track of the lane and warn you; sensors to monitor driver alertness; sensors to avoid hitting the preceding car. There's a lack of pedestrians and cyclists, and the rare crossing animals are best handled by a processing unit which does not have a 1000ms lag (or longer, if tired) or which cannot see sideways in the darkness. The consequence of an incapable driver is very bad (due to high speeds), while the technical feasibility is close (e.g. the safe stopping in case of the driver having an attack). Also, the transfer of control can be well-timed; it's unlikely that a city just pops up suddenly. So my bet is highways rather than HOV lanes. It is unlikely that the initial version will have convoy forming abilities, or that new lanes will be added for the autonomous cars exclusively. We need the network effect, and changing cars is easier than changing infrastructure. First there will be lots of cars that can drive themselves, then infrastructure will slowly adapt.
Alternative road types: self-parking cars in dense urban plazas, hotels etc., replacing valet services in controlled environments. Or war zones for transport (where it is probably happening by now) - essentially no liability issues. Or restricted areas of urban traffic (e.g. areas for the use of public transport only).
Type of vehicle: it may not be passenger cars initially. Maybe utility vehicles that slowly traverse long segments of the highway in the emergency lane, or trucks (which are in many countries have limited top speed and may be prohibited from overtaking vehicles or use any of the lanes except the outermost.
Alternative use case: maybe a sufficiently advanced car will be indistinguishable from a self-driving car. I.e. the person is legally driving the car, and the car - at many levels of planning and abstraction - modifies the driver's actions. ABS, traction control and engine controls already provide lower level overrides. For example, initial versions of higher level utilities may beep if you were to cross into the other lane while there is a car in the blind area. A next version may steer back. Volvo has some technology to measure car distances ahead and apply brakes if needed. So a drunk person may sit in the car and, importantly, the rest of the society would become somewhat more protected from this guy. He just pushes the gas pedal and drives erratically, while his car mitigates his actions, keeping him in lane or braking for obstacles as needed. Navigation technology may also become multimodal: it may gently steer the wheel toward the lane you'll need, just as a signal. Such things would be more assertive if there was some sign that the driver became asleep or out of control. So in some sense, pushing the gas pedal stops moving the rod that opens the engine air inlet (as it did in the past) and starts signaling the intention of the driver to go ahead.
It is more likely that cars will be eventually mandated to take control from humans in case of an immediate emergency, rather than the other way around, so there is no need to worry about returning control to man. Humans do as well as they can but the responses are very poor in emergency situations. Since it is not practised by drivers, the outcome is very stochastic. There is lack of attention, long reaction time, incorrectly applied brake, oversteering etc. On the other hand, cars may have ever growing situational awareness: cars may eventually form ad-hoc networks (including human-driven cars), sending speficic notifications; their sensors and reaction times are superior; their attention constant; their strategies communicable.
Technically there is a very small difference between a very safe, intelligent car that assumes/overrides controls as needed, vs. a fully autonomous vehicle. The really interesting question (besides the technology) is how it will change society, habits, urban landscape and interactions.
Stopping a car while a man is having a heart attack is going to be an amazing thing, that I think you are basing this on performance characteristics that do not exist and are not reasonable.
I recommend watching videos like DARPA urban challenge and Google's self-driving cars, which racked in hundreds of thousands of autonomous driving with very occasional intervention in highway and urban settings. You can try and understand a bit of how they work, including how they combine sensor data, detect obstacles and form plans on the go. Can they be tricked? Yes. Can they be more effective than an uncapable (inattentive or shocked) driver in making choices or bringing the car to safety? Most probaby, we are about there now. Are they bound to be superior in all important regards of driving safety? Absolutely. There is no such a thing as THE fundamental thing to solve, and it's not clear what you mean. It's lots of small, tedious steps that usher in this "manifest destiny".