As PM 2Ring answers below, while choosing the point to be at $\mathbf{x}$ wrt. the spaceship would result in acceleration toward $\mathbf{x}$, choosing $-\mathbf{x}$ will result in the opposite, so they should cancel and hence result in no acceleration. But once some $\mathbf{x}$ is chosen, then all mass in the universe is accounted for exactly once, so there's nothing left to cancel any acceleration.

However, in my (obviously flawed) thought experiment, I can at any point in time reconsider my sphere and shells to be centered at $-\mathbf{x}$ and make the ship accelerate in the opposite direction — just not at the same time.

Of course this doesn't work, but why?.

Of course this doesn't work, but why?

1. The gravitational attraction of a spherical body acts as if all its mass were concentrated at the center, and

2. The gravitational acceleration inside the cavity of a symmetry body vanishes.

1. The gravitational attraction of a spherically symmetric body acts as if all its mass were concentrated at the center, and

2. The gravitational acceleration inside the cavity of a spherically symmetric body vanishes.