They used the cheaper servos for this build, understandably, but if you buy some more expensive ones you can probably make it nearly silent. This would also likely let you make the servos move faster and reduce the time it takes to update (and thus make noise) at the same time. I'd also consider maybe using solenoids to move the magnets in and out which could be completely silent then too, at the expense of needing some more current when engaging if you add some damping, just a little felt at the end of the rod. Then switch to steppers for the rotation and lifting the ball platform and you could likely have it completely silent.

You will quickly learn to live with it, just like with older mechanical (granfather's) clocks.