Correlation of entangled states under unitary evolution

Context: I am a researcher in GR, but much more recently self-taught at QFT etc, and trying to improve my understanding.

Thought experiment:

An entangled pair of particles is created. Their ‘initial entanglement’ means that their ‘initial conditions’ (say, their spin directions at the start) are unknown, but perfectly correlated. Then, the particles are separated, and kept away from ‘noise’ etc, so the entanglement is preserved.

While they are being ‘held apart’, before anything interacts with them, they are presumably undergoing unitary evolution, or at least this is my understanding of QFT – i.e., the Hamiltonian is used to evolve each state forward in time from t_1 to t_2.

Unitary evolution is given by a 1st order differential equation, and therefore if we have uniqueness of solutions (determinism?) then surely, correlated initial conditions naturally imply that states will remain correlated over time…

My question is: Given this, why is entanglement so surprising, or viewed as being "non-local"? Or, more likely – what error am I making? 🙂

Thanks for any help and advice!