Your point illustrates the subject line nicely. Very many people have a disability, yet you're suggesting very rare benefits of keeping kerbs that hamper people with disabilities.
(Yes, riding on sidewalks is technically illegal. There are many places where it's much much safer and there are no pedestrians to bother.)
The reason is simple: Bikes move quickly, and bikes on the sidewalk are much harder for motorists to see than bikes on the street. That makes it extremely difficult - sometimes impossible - for drivers to see or anticipate when a bike is about to cross a street, alley, or driveway. Meanwhile, cyclists are relatively unaware of their surroundings compared to pedestrians, for the same reason that motorists are: They're moving faster. These two factors can easily combine to create a sudden use-of-space conflict that cyclists never win.
That differs by jurisdiction; in many places it's completely legal.
It's far, far safer to be on the sidewalk where I live.
I live in a city known for having many cyclists, and there are always tensions between the local cycling population and the authorities about the poor level of law enforcement against dangerous drivers. Of course, there is also a poor level of law enforcement against cyclists doing dumb and illegal things. The difference is that when something goes wrong, it's usually the cyclist who ends up injured or dead, not the motorist.
So while in an ideal world we'd all get along nicely and play by the rules, I don't tend to get angry at cyclists whose choices are illegal but obviously safer (as in, I've read the actual research and current law and/or public policy clearly doesn't promote the safest option, not just doing something that intuitively feels safer but maybe isn't really).
In the case of riding on sidewalks where it is inappropriate, it's also frequently a pedestrian.
"I don't tend to get angry at cyclists whose choices are illegal but obviously safer (as in, I've read the actual research and current law and/or public policy clearly doesn't promote the safest option, not just doing something that intuitively feels safer but maybe isn't really)."
In those cases, I agree that the cyclist should do what's safest (presuming it's safe enough - on balance - for everyone involved). So should drivers of vehicles, for that matter. My understanding is that taking the lane, where traffic is moving at a speed the cyclist can keep up with, is the best way of keeping everyone safe (I wouldn't be recommending it otherwise). You are right where everyone who might hit you can best see you. I say this as someone who cycles in a big city fairly regularly, with a spouse who bikes in a big city even more regularly, in the wake of biking safety classes. I'm not just an annoyed driver/pedestrian/whatever trying to keep the cyclist down.
A bicyclists safety is much more threatened by a multi-thousand pound vehicle traveling at 40 or more mph with a distracted driver on a seemingly deserted stretch of road, then a pedestrians is by a 150-200 lb body traveling at around 10-15 mph. The reaction time is also much greater, allowing more opportunity to prevent a collision on the part of the oncoming vehicle.
Of course, there are still situations where riding in the street - either in a bike lane or taking the lane - is dangerous or otherwise impractical; the question is whether riding on the sidewalk is an improvement, or whether you should be forgoing cycling (perhaps walking your bike).