Suppose I have a source that produces pairs of spin-1/2 spins in the entangled state
$$ \frac{1}{\sqrt{2}} \left( \mid\uparrow_z\uparrow_z\rangle + \mid\downarrow_z\downarrow_z\rangle \right). $$
The particles are sent opposite ways into respective Stern-Gerlach (SG) detectors where the spins are measured. One of the particles is measured first (doesn't matter which one).
If the Stern-Gerlach representing the first measurement is oriented vertically, to align with the z-axis, then the measurement (meaning the detection of the first particle after it passes through) causes the particle to assume one of the two states $\mid\uparrow_z\rangle$ or $\mid\downarrow_z\rangle$. This in turn causes the other particle to assume the same state, and therefore both particles come out of the same channel of their respective Stern-Gerlach setups. I'm on board with this.
Now suppose we turn both Stern-Gerlachs to a new angle. The same thing will happen, meaning the two particles will always come out in the same channel. I have an image in my head of an SG measurement effectively disturbing the particle and destroying the old information we (might have) had while forcing the spin to pick a state in the new measurement basis defined by the SG's orientation. So how can the correlations survive a measurement in a basis other than z? What is the intuition for this? How can I understand intuitively why being entangled in the z-direction means that measuring one particle at some angle $\theta$ causes both particles to collapse to $\mid\uparrow_\theta\rangle$ (for example)?
Happy to edit or answer clarifying questions.