Why does the Higgs field fall into the same ground state at all points across space?

If I understand it correct, then in a nutshell, you are asking why is the VEV independent of spacetime. If the Higgs field had different values at different points in space i.e., if it had a spacetime variation, then the gradient term would give a positive contribution to the Hamiltonian, and hence, the total energy will not be minimized.

The OP asks:

I understand that the circle of degenerate minima form a spherical shell $\phi^{\dagger}\phi=\frac{\nu^2}{2}$, so why does the Higgs field choose the same point on this shell across all space?

The shape of the potential that gives rise to the degenerate minima, forms this spherical shell as a collection of continuously connected points that all give the same VEV. So, to say this differently, the value of the VEV would not have been different if another point on the shell has been chosen. So then why this particular point?

Actually, one should perhaps not assume that the point on the shell is the same throughout space. We already know that Goldstone bosons represent excitations of the motion along the valley, in other words, the motion of the point as it remains on the shell. So, in principle the point could be moving around on the shell as we move from point to point through space. However, as long as it remains on the shell, the VEV would be the same everywhere.

Important question!

When the Higgs symmetry breaks to give a particular vev at the first point $x_1$ then, as other answers point out, neighbouring points will break to the same value. The domain will spread, presumably at lightspeed.

But in a large universe there may well be some point $x_2$, far from $x_1$, which chooses a different vev, and will start its own spreading domain. In time the regions will meet and form a domain wall. Like a ferromagnet.

(I have more than once sat at a circular table at a conference dinner where guests had to choose whether to use the glass on their left or on their right. Usually one brave person would make a choice, their neighbours would follow, and all would be well. Sometimes two brave people, some distance apart, made different choices so that one guest finally ended with two glasses and one with none: they were the domain walls.)

As the observed universe all has the same Higgs vev (as far as we can see) that means it must have been causally connected when the symmetry was broken. So this is a big argument for inflation. The Higgs symmetry was broken with different vevs in different regions and all sorts of interesting topological features, but an enormous expansion took a small homogenous patch and blew it up to the scale of our observed universe.