Can perfect turn coordination automation be achieved by linking rudder to aileron?
The mechanical linkage from control column (aileron) only addresses one of the things that affect coordination, the adverse yaw. It is the most significant one, but not the only. Other reasons you may need some rudder deflection in coordinated flight include:


Compensating for P-factor. At high angle of attack (slow flight) the propeller blade going up sees a lower angle of attack than the propeller blade going down, creating asymmetry of thrust. The effect depends on engine power and speed, making it too complex to compensate by just mechanical linkages.
The rotating propeller slipstream also creates asymmetrical force on the rudder, requiring compensation depending on engine power and speed.
In tail-draggers, the propeller precession creates rather significant yawing moment as the tail lifts off that needs to be connected using rudder, otherwise the plane could ground-loop. Tail-draggers generally require active control by rudder, because they are directionally unstable on the ground.
In multi-engine aircraft, asymmetric thrust in the event of engine failure. For multi-engine aircraft this is the most significant yawing moment that the rudder must be able to compensate.
And last but not least, rudder is needed to damp the Dutch roll. I've never heard that mentioned as problem in utility aircraft, but it is a big problem for transport aircraft and a reason why they usually come with a yaw damper.


The yaw damper is a feedback controller that monitors the lateral acceleration and other parameters (yaw and roll rate) and applies rudder to damp the dutch roll, and also to provide coordination to turns flown by autopilot where usually the autopilot directly commands the ailerons and rudder is left up the the yaw damper. So yes, providing coordination is possible, but not via simple mechanical linkages. And of course the rudder pedals still need to be there so uncoordinated flight is possible for any of the reasons already listed in Ron Beyer's answer.