OK, thinly disguised but simplified Spinlaunch physics question-
Assume all in vacuum, no gravity to speak of, classic 'assume the cow is frictionless' conditions.
Ball at end of tether, tether attached to an axle at other end, axle is spinning, ball is describing a circle, tether releases, ball goes off on a tangential path.
Does the ball spin after release?
Almost certainly yes. If the rotor at rest holds the ball rigidly so that it maintains a fixed orientation, the ball will will rotate once per revolution as the rotor spins. The rotor spins up to 450 rpm (7.5 revolutions per sec), and the ball rotates at that rate. It will keep that rotation when released. In the scheme, the centrifugal force points to the same side of the ball as it spins up.
You could hold the ball with a hinge, so that the ball maintains a fixed orientation with respect to the earth during spin up. In that case, the ball would not rotate, and all would be well after launch. The ball would feel a centrifugal force that always points away from the center of rotation. If the ball doesn't rotate, the ball feels the centrifugal force rotating around it once per revolution. This might make it harder to design a vehicle that can stand up to launch forces.
Suppose the hinge doesn't line up perfectly with the center of mass of the ball. Centrifugal force would try to orient the ball, which would give it spin.
I can only guess that the true launch vehicle does spin, and that it has an aerodynamic shape like an arrow. Perhaps wind forces straighten it out after launch.
Update
Following the links in the comments to How much energy is lost by damping yaw from a SpinLaunch? from Space SE gives the answer.
The launch vehicle is long and attached to the rotor at multiple points.
As each attachment point becomes level with the center of rotation, that point is going exactly up.
As the nose passes the level point, it is detached. The nose is now going exactly upward and has no centrifugal force pulling it sideways. The tail still has an outward component of velocity and a large centrifugal force decelerating that component.
When the tail passes the level point, it is detached. The nose is still going straight up, and now the tail is too.
Leaving the tail attached slightly longer has applied a large torque that slows the rotation of the launch vehicle to a stop.
In practice, there will be multiple attachment points. Applying that much torque at just the tail creates a lot of stress.
As a simplified explanation, you can observe the movement of the ball at the point when it is released: Since it is rotating with the tether, a point on its "far" side (more off-center) is moving slightly faster than one on the "near" side.
This difference is what causes the rotation -- if the ball didn't "compensate" the speed difference by rotating, it would need to do something harsher (such as disintegrating).
