The mass isn't rotating as it exits the launcher.
Edit: Here's a link to an illustration, as your link provides zero information in that regard.
https://www.youtube.com/watch?v=TGO4LtCctTk&t=141s&ab_channe...
I clearly see their rocket rotate around its center of gravity, with the same angular velocity as the arm it's sitting on, while the center of gravity is moving on a circular path.
So, again, how is it not spinning? Where does the angular momentum go?
The launcher does not spin the payload around the long axis.
It is indeed exactly like a bucket. And the bucket changes its orientation throughout you spinning it, that's the whole thing, that's why the water stays inside. The top side of the bucket is always facing you. For that to happen, it has to rotate around its COG with the same rate it's rotating around.
A different example: the moon is tidally locked to the earth (like the bucket to you, or the rocket to the spin launcher) - very obviously, it needs to spin around its axis to achieve that, and it does so at exactly the same rate as it is rotating around the earth.
That momentum doesn't just disappear.