There are some things that bias what happens at the start, though. We aren't firing it in space where everything is free to move in any direction without bias. We're firing it on the surface of the earth, in its gravity, with it oriented in a specific way. A bare barrel, i.e. sans tuner, isn't stiff enough to not sag by some amount under its own weight. Adding a tuner in cases where you'd use a tuner adds to that sag. In both cases, that biases how it will move once it starts moving. And it is biased in the same plane with regard to the overall rifle's centre of gravity, too. You don't tend to have a lot of mass hanging off the side of a rifle. What does hang off of it tends to be in the vertical plane. Trigger and stock below, rings and scope above. So that, too, will bias the movement to some degree. Does the barrel to exactly the same thing every single time? Of course not. But its general movement is similar enough from shot to shot that we can consider it a predictable thing. I think it is safe to say that the variance in the ammo is a larger source of variance on target than the movement of the barrel.
Kolbe's barrel movement traces don't show indications of random initial movement direction. Nor do typical shooting results. If it were randomly flipping the up/down orientation of the initial barrel movement that would affect the rest of the movement trace accordingly, and you'd see wild fluctuations in vertical on target as a result. A well-tuned gun that puts round after round through the same ragged hole would be impossible. If bullets are generally leaving the muzzle of a given barrel with a 0.25-0.5 MOA addition to the initial launch angle that would mean in half the cases where it started moving in the opposite direction you'd see it swinging 0.25-0.5 MOA downwards instead. That would mean 0.5-1.0 MOA variance on target due to nothing but the change in launch angle. Add in the effect the muzzle velocity would add to that and you're talking about even more vertical on target. A slow shot leaving at +0.5 MOA launch angle when the barrel is swinging upwards would then be leaving at -0.5 MOA launch angle when the barrel is swinging downwards. So not only is it already going slower, it is now aimed 1 MOA lower than before, so it is going to hit even lower. You'd never see ARA benchrest shooters shooting perfect 2500 scores if that were happening. It would be impossible. No, I think it is safe to say that barrels move in generally the same fashion from shot to shot. Not perfect molecule-for-molecule repeatability, of course. But more or less the same way every time. The ammo's own component variance surely outweighs it by a good deal.