The benefit of using tapers is that all the clearance is removed, which is very important when rotating the workpiece (WP) in precise 90 degree increments. The cost is to ensure the design provides an axial force that exceeds the cam-out axial force generated by the pin. I used a 5 degree taper angle in my example. A 2 degree taper would generate less axial force but the pockets might need to be deeper and retract more to accept a longer pin, which would be needed to maintain the same misalignment tolerance to get the pin in the hole. As long as the linear actuators on the ends can generate a force that exceeds the axial reaction force of the pins, that is an acceptable condition to have in the design.
There are collars that clamp onto a cylindrical shaft, which when tightened, also have a clearance-free connection to the shaft. That is more difficult to implement in a machine that is being designed for automated WP loading, which is an objective of the current design.
I'm also thinking about the fact that the Center of Gravity (CG) of the WP is off the axis of rotation. I suggest you characterize the rotational stiffness of the interface between the features on the shaft end and the receiving features on the end of the machine that has the motor that rotates the WP. It looks like the opposite end of the machine has just a bearing to allow free rotation. That means all the torque generated by the radius of the CG from the shaft axis must be reacted by the features at one end only. If you know the stiffness of the interface, you can apply the maximum moment and determine the angle of rotation of the shaft. Now look at the maximum distance on the WP from the axis to see if the displacement of that point on the WP has an acceptable deviation due to the flexibility of the coupling. If not, you will need to look for an alternative coupling design that would have much higher angular stiffness than a tetrahedral pin in a tetrahedral hole.