The car approaches blind corners at a safe speed and at a great enough distance to give the car and oncoming cyclist adequate time to react.
There are several solutions to this. The first would be to switch to airless tyres, but they're still prototypes at the moment.
The second is to use road sensors to avoid sharp debris and use embedded sensors in the tyre wall as well as other suspension-based metrics to monitor tyre health, and pull the vehicle out of service when those metrics reach unsafe levels
Finally, the car should model blowouts and the reaction to them on different road grades and conditions so that it can orchestrate a predicted response to a blowout that couldn't otherwise be detected (like a lucky gunshot, I guess).
But we're talking about automated vehicles, so the car brought itself into the situation. Logic follows that an automated car will always avoid scenarios it cannot exercise adequate control over. Automated cars must, by necessity, be incredibly paranoid about both their own state and the state of their surroundings.
Good luck convincing people to buy cars they know will murder them.
Events that include complete loss of motor control are somewhat out of scope. If a bridge collapses under any vehicle, there is precious little it can do than offer as much protection to its occupants. Obviously, an automated car could try and find a less risky route...
Any automated car that ends up killing someone in a situation it had control over is buggy.
The car detects the obstacle and slows down on approach. The car then moves past the obstacle at a speed it is able to cope with any further changes to the environment.