I can tell you exactly where it came from, it's being spun up in that launcher thing, obviously, that's the whole point of spin launch. You can see it for yourself in the video. Is the pointy end of the rocket always facing in the same direction, e.g. upwards? No, it's changing its orientation continuously, it's always facing 90 degrees from the arm of the launcher, which, you guessed it, spins.
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.
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.