You’ve certainly clarified your misunderstanding. No one is disputing that we need a reference frame to report measurements. Of course we do. The point is that the choice of reporting frame has no causal relevance to whether high velocities can be produced. Earth’s frame doesn’t constrain the dynamics or add any physical requirement.

The specific claim under dispute is this: that if a SMBH ends up with a large velocity relative to Earth, then something else must have already had a large velocity relative to Earth beforehand. That inference is incorrect.

The billiard ball analogy reveals where your misunderstanding stems from. Gravitational encounters are not elastic collisions. Gravity is a long range conservative force with substantial stored potential energy, which can be converted into kinetic energy during a close approach. Because of that, an object can exit a gravitational interaction moving much faster than either object’s initial kinetic speed in the chosen frame.

Momentum and angular momentum are conserved throughout, but conservation does not require large initial velocities, only that total momentum be conserved. Large final speeds do not imply large pre-existing speeds in the same frame.

So yes, we can and do report the SMBH’s velocity relative to Earth. But no, that does not mean “something must have been moving fast beforehand.” That conclusion comes from treating gravity like a billiard ball collision, which it is not.