Most aircraft have the stall sensors for warnings, right down to small private planes.

The 737 Max 8 is different because the MCAS system actually overrides the pilot's input, rather than just sounding a warning to alert the pilot to a potentially dangerous situation.

Airbus and Boeing take different approaches towards flight control. Airbus will override pilot inputs to keep the plane from stalling or otherwise flying out of the envelope. Boeing allows pilots to override the plane even if the plane thinks it's a dangerous maneuver. Airbus's system needs to be more reliable since a malfunctioning sensor will cause the plane to crash itself.

Wasn't the problem that Boeing requires pilots to keep overriding the system in the case of a failure, while not training them to do so, or even mentioning the fact in the documentation?

Yes, but the theory was that since the pilot could override the system it was less critical compared with Airbus where the systems controlled by the sensors are considered critical for flight. That's why Boeing used 1 while Airbus uses 3.

Do you know exactly how fly-by-wire Airbus planes are? In an emergency, if the avionics computer goes crazy and starts making the plane do barrel rolls, can the pilot override it and take manual control?

>Do you know exactly how fly-by-wire Airbus planes are?

In normal operation it takes the input from the pilot and decides how to accomplish what it thinks the pilot wants. For example, if the pilot pulls all the way back the plane will translate that into the maximum safe climb rate. You can think of it as the pilot expressing a desired state and the plane figuring out how to get there.

>In an emergency, if the avionics computer goes crazy and starts making the plane do barrel rolls, can the pilot override it and take manual control?

AFAIK there's no way to disable the flight control. The computer is supposed to be reliable enough to not do crazy stuff and it accounts for sensor errors. Based on what sensors are malfunctioning different safety features are disabled. For example, if it can't get a reliable angle of attack reading the stall prevention is disabled.

Airbus systems have alternate law mode which kicks in when computers, sensors fail, read about Air France crash

I think this is true. Also, in the Air Asia crash when pilot disengaged autopilot by mistake using a reset switch changing flight mode to manual.

As far as I know, if the autopilot can't make reliable decisions, it rings alarm bells and disengages fly-by-wire. Looks to me like quite elegant design.

I see, though for the airbus I'd have thought a warning system would be better to implement than more complicated to implement control system.

Many planes have angle of attack sensors. Normally it is just a warning system. Boeing used theirs as input for their control loop of the MCAS system.

And most have redundant systems.