The answer is, that there are forces/torques you are overlooking.

Suppose, looking directly down from above, we focus on a teacup to the "south" (bottom). Quarter of a rotation of the entire ride later, its now to the "west" (left) instead. So the cups axis of rotation itself, has shifted.

That took place because of forces at the point it's mounted to the ride. Your summary overlooks those.

Suppose the original teacup, at the "south", had an axis tilted from bottom left to top right (rising west to east). When the ride turns quarter of a turn, that teacup now has an axis rising from bottom-south to top-north, and different rotation, as well as the teacup itself having a northerly velocity vs a westerly one.

All of those changes occurred because the ride can force the mount-point in the given direction, and the mountpoint can transmit that change to the teacup, with more than enough force that the teacup changes its movement and axis, if you want an informal explanation. Because reactions have an equal and opposite reaction (ride overall isn't changing position etc), there will be an equal and opposite set of forces and torques on the base of the ride - but the opposite position teacups probably balance each other out, leaving little net force/torque.

If you want a simple but not exactly the same parallel example, when you put a block on a rough level surface and push it very lightly, it doesn't move. The reason is that friction and reaction forces occurs at the junction, and unless you take that into account, it won't make sense when you analyse it.

The answer is, that there are forces/torques you are overlooking.

Suppose, looking directly down from above, we focus on a teacup to the "south" (bottom). Quarter of a rotation of the entire ride later, its now to the "west" (left) instead. So the cups axis of rotation itself, has shifted.

That took place because of forces at the point it's mounted to the ride. Your summary overlooks those.

Suppose the original teacup, at the "south", had an axis tilted from bottom left to top right (rising west to east). When the ride turns quarter of a turn, that teacup now has an axis rising from bottom-south to top-north, and different rotation, as well as the teacup itself having a northerly velocity vs a westerly one.

All of those changes occurred because the ride can force the mount-point in the given direction, and the mountpoint can transmit that change to the teacup, with more than enough force that the teacup changes its movement and axis, if you want an informal explanation. Because reactions have an equal and opposite reaction, there will be an equal and opposite set of forces and torques on the base of the ride - but the opposite position teacups probably balance each other out, leaving little net force/torque (ride overall isn't changing position etc).

If you want a simple but not exactly the same parallel example, when you put a block on a rough level surface and push it very lightly, it doesn't move. The reason is that friction and reaction forces occurs at the junction, and unless you take that into account, it won't make sense when you analyse it.