Can the narrative of spooky action at a distance be derived from the mathematics of quantum mechanics?

The usual popular (and also not so popular) science lore about entanglement goes about like this:

Two particles enter into contact (or a single particle decays into two new particles) and hence get entangled with each other. We bring the two particles far away in space without touching them. Then we decide to make a measurement on one of the two particles. Two things then happen:

1. Both particles fall in a definite state in the same instant.
2. If we repeat the experience many times (with other particles prepared in the same way) we find out that there are statistical correlations between the states of the two particles (just after measurement) that cannot be explained classically.

On the other hand, I know (at least as a proof of concept, modulo details that I have never seen written down entirely, but I’m willing to believe) that what happens is the following.

We can factorize the Hilbert space of the two particle system as a product of one-particle systems; the initial state $psi_0otimesvarphi_0$ is separable, and unitary evolution transforms it into a non-separable (aka entangled) state
$$U(t)(psiotimesvarphi)=psi_1otimesvarphi_1+psi_1otimesvarphi_1;.$$

Question: how can we r̶e̶c̶o̶n̶c̶i̶l̶e̶ derive the popular science narrative w̶i̶t̶h̶ from the actual mathematics? (i.e. by the abstract postulates of QM plus maybe a more specific concrete description in position representation)

(I’m ok if the answer skips on the "non classicality" of correlations part, if it amounts just to the Bell’s inequality, cause explanations of it can be found in countless sources)