I think it is a mistake to cede the roads to motorists. Such a mistake give cars a false sense of permanent entitlement over anything but another car.
The truth is that everyone has the right to use the public roads.
I am not sure that there is any need for even being concerned about anybody needing to cede anything.
Bicycles have. as you point out, a legal right to be on most roads (they are properly forbidden on some roads such as freeways) in the exact same way that the horse and buggies that one sees in areas where there are Amish settlements have a right to be on the road. If anything, recognition of the rights of bicyclists to be on the road has been on the uptick in recent years - for example, the installation of bike lanes and Google Maps providing bike routes.
My only real concern in this area is one of safety. People who are not trained, skilled or prepared for driving in traffic should not attempt to do so until they acquire such skills and preparation. And to encourage such people to ride in traffic is highly irresponsible and could result in someone getting seriously hurt or worse.
At the most fundamental level, I think the biggest problem is a certain commonplace mindset among the general population that are not avid bikers that thinks of a bicycle as being a benign toy. And, if you stop and think about it, this mindset has long and deep roots. It is almost a rite of passage in suburbia - a child starts out riding a tricycle and, as he or she gets older, graduates to a bicycle with training wheels and, from there, to a child-sized bicycle when the child is still in elementary school. That is how most people in the USA are exposed to bicycles - they are something that is part of childhood fun and recreation and not so much a grown-up world utilitarian object.
My own particular childhood gave me a first hand exposure to a very different perspective. My mother is British and did not learn how to drive until she was about 33 years old. While growing up in England and spending her early adult years in Europe she got around exclusively by walking, transit and, yes, sometimes by bicycle. One can do that over there. (My English grandparents lived into their 90s and never owned nor had any need to own an automobile).
And the culture clash between England and the USA when she came over was much greater than it would be for someone today. The decades immediately following World War II were extremely difficult ones for the UK and many aspects of daily life in parts of the country remained in a sort of time warp compared to life in the USA.
For example, my mother never saw a supermarket until she was an adult - people still had to buy meat, bread, fish and produce in separate standalone markets. People in her town lived in row houses and their "back yards" were fenced in squares of pavement about the size of a small patio and contained a shed for coal and separate building with a Victorian era type of flush toilet that was powered by recycled waste water from the kitchen sink. My grandparents' house was more luxurious than many others in that it had a permanent bathtub with water heated by a back boiler from the kitchen coal stove. Other houses had to use portable bathtubs with water heated by kettles and it was considered bad form on one night of the week (Saturday, if I recall what she told me correctly) to call on people as it was assumed that the family would be busy taking their baths. People in her part of England lived like that well into the 1960s when things gradually began to modernize. (And that made for an interesting childhood for me as, like many other children of immigrants, I was exposed to different cultural perspectives at home and at school - something I now consider to have been an advantage as it taught me that so-called "conventional wisdom" is not sacrosanct. and that other perspectives are possible).
As a new driver who only learned relatively late in life, my mother was, understandably a cautious driver - it took a few years after she started driving to feel confident enough to begin driving on the freeway. One of the things I remember that stressed her out to no end was driving through residential neighborhoods when kids were playing near the street. Sometimes the kids would suddenly dart out into the street - often on a bicycle - without looking. I vividly remember a few instances where, despite the fact that she drove slowly, she had to slam on the breaks to avoid hitting a kid.
This was something that really frustrated her. And I remember a common rant of hers was about the unsafe manner in which American kids used their bicycles. A bike is not a toy but rather a form of transportation she would say. She decried American parents for not teaching their children how to use a bike safely and responsibly. She stated that bicycles are required to follow the same traffic laws that cars are which includes using (hand) signals when making a turn. She would state that if a child behaved in a similar manner on a bicycle when she was growing up it would have received an immediate scolding from any nearby adult - and the post 1960s American taboo against scolding other people's children was also foreign to her.
This was all part of her wider culture shock at what she felt was the irresponsible and lax way that American parents raised their kids. She was appalled at how parents would commonly give in to their children's tantrums and she strongly disapproved the degree to which American parents allowed their children to consume so many sweets, sodas and junk food. Michelle Obama and Michael Bloomberg's speech writers could have just taken her rants from that era on the subject and have saved themselves a lot of work. In our house such things were served only as an occasional treat or on special occasions.
I remember myself thinking at the time whenever she would go on about such matters "yeah - but in America things are different." Years later, as an adult, I realize, however, that she was spot-on right - both on the account of the bikes and on American dietary habits. On the other hand - in defense of the era's suburbanite Texans - at that time in suburban Texas there was very little basis for people to think of bicycles in terms of being a serious form of transportation. A bike is pretty much useless as a practical transportation option in the sprawl of suburbia. And the notion of somebody in that era wanting to live in an urban area, let alone ride a bike in one, would have been regarded as odd and downright radical. People who could afford to do so were fleeing the center cities which were largely becoming run down and properties were being converted to low rent uses or just empty lots. And the reason the children my mother grew up with were instilled with such healthy eating habits was because sweets, when they were even available, were expensive as war time sugar rationing in the UK was not lifted until 1953. In that time and place, healthy food and cheap food tended to be one and the same. In the USA the exact opposite was and is true - thus eating healthy requires much more conscious intent.
Anyhow, I think the suburban mindset in our culture of a bike as a childhood toy associated with fun and recreation is a problem in the context of what we have been discussing and it is potentially dangerous if that is the only exposure one has had to riding a bike and one suddenly decides to start using one on dangerous and busy city streets.
Just because one has the skill and ability to make a bicycle go from point A to point B does not mean that the person is prepared to ride in city traffic. Riding a bike through the streets of a sleepy suburban neighborhood is not the same as riding in city traffic. If people think that it is and they decide to venture out in city traffic, they are putting themselves at needless and dangerous risk - and it is irresponsible to encourage such people to do so. And the safety tips on the baskets of the rent-a-bikes hardy constitutes serious training.
If you are not a very skilled or capable automobile driver, you will NOT be a good candidate for riding a bike in traffic. If you frequently catch yourself mentally drifting while driving a car you will NOT be a good candidate for riding a bike as the need for sustained focus is greater and the built in margin for error is much less.
It is no different than if some group went around encouraging the general public to just run out and buy a handgun and the city came up with some sort of multi year plan to make it easier for people to do so. Yes, there are people for whom it makes sense to own a hand gun for personal protection. Yes, it might be wise to encourage certain individuals facing particular circumstances (for example, a deranged, violent and vengeful ex) to get a gun to protect themselves and their families. But even here it would be highly irresponsible to encourage a person to do so if part and parcel of "getting a gun" does not include learning how to safely use, handle and store it. And there are some people one should NOT properly encourage to buy a gun - people who have anger management issues, people who are careless in pretty much everything else they do, etc.
Just as there is more to owning a gun than point, aim and fire there is more to riding a bike in traffic than just getting on the thing and peddling. And just as it is a bad idea for some people to own a gun, it is also a bad idea for some people to ride a bike in traffic.
If some sort of gun advocacy group decides that it would be beneficial for there to be more gun owners on grounds that having more people in their ranks will give themselves greater political clout when it comes to gun related issues or on grounds of some sort of wider ideological agenda - that still wouldn't make it any less irresponsible for them to go out and just encourage as many people as possible to go out and buy a gun.
I can understand why environmental and certain bicycle related activist groups would like to see more bikes on the road. More people riding bikes will give them greater political lobbying power and people who have ideological agendas have every right in the world to promote and push those agendas. But none of that excuses encouraging people who are not safely prepared to ride bicycles in city traffic to just go out and do so. The ends do not justify the means.