Why don't they have yaw dampers if they are always "suffering" from adverse yaw?
A yaw damper means added cost, weight, complexity, and one more system that has to be maintained. That's the downside. The upside, for most GA aircraft is that they relieve the pilot of having to make a "coordinated" turn. In other words, having to apply appropriate rudder input along with aileron input. Is that convenience worth the downside when the pilot has internalized doing that without having to think about it? That's a matter of opinion. For myself, a yaw damper in any of the, say, single engine Cessnas would be a waste.
There are some GA aircraft that could benefit from a yaw damper in other than banking into a turn or leveling from a turn. For example, in level flight the old V-tail Bonanza would yaw around the vertical axis in turbulence to an objectionable degree in my opinion. Very uncomfortable, especially to passengers in the back seat. It would have been great to have had a yaw damper.
Adverse yaw is the yawing moment that occurs due to aileron deflection.
- Yawing moment from aileron deflection: upon a left turn the right aileron deflects downwards and the left one upwards. The downwards deflecting aileron creates more drag friction, pulling the nose of the aeroplane to the right.
- Yawing moment from roll velocity. The wing rolling up has a different Angle of Attack than the wing rolling down: the downwards going wing has the lift vector tilted forwards (like in a glider) and vice versa, which creates a yawing moment pulling the aeroplane nose to the right.
So adverse yaw pulls the nose into the opposite direction of where we want to turn, and the effect scales with wing span. One way to counteract this is to couple the rudder to the aileron, and give automatic nose left rudder deflection upon left aileron roll. Some of the Learjets and I'm sure some other aeroplanes have this, a way to automatically fly coordinated turns.
Yaw dampers on smaller planes do not coordinate turns but damp out Dutch rolling: the rudder deflects to counteract yaw velocity. Only airliners have yaw dampers that also coordinate turns.
The thing is that GA aircraft are mostly manually flown, control surfaces are manually deflected. For a yaw damper you need an actuator, so will need to install a powered rudder system. A major cost driver.