Why is quantum non-locality so surprising when quantum mechanics is a non-relativistic theory to begin with? Ignore the fact that Quantum Field Theory (QFT) is compatible with Special Relativity (SR) for a second. Imagine we are back in a time before the construction of QFT. If Quantum Mechanics (QM) is the quantization of a classical non-relativistic theory, why should we be surprised by things like non-locality (specifically from entanglement)? I recall in my QFT for condensed matter class, we even had instantaneous Coulomb potentials, which manifested as non-local vertices in Feynman diagrams. So non-locality is not an uncommon thing in QM, right?
So why was Einstein so initially bothered by "spooky action at a distance"? Shouldn't one be more concerned if this kind of thing shows up in relativistic QFT? (which I assume it does, but it feels improper to discuss non-locality in a non-relativistic setting)
For clarity, when I say QM here, I specifically mean non-relativistic QM.
 A: There's two layers of "nonlocality" at play here.
On the one hand, there's the issue of "action at a distance". Relativity tells us that you cannot have something affect something else faster than the speed of light. Note that here "affect" is to be understood as something along the lines of "there is no way to measure a system B and find that its state has been altered due to something that happened at a system A outside of its light cone".
On the other hand, quantum mechanics can display "nonlocal effects" of a different nature. Namely, there can be correlations between distant observers that cannot be explained via any kind of "shared classical correlation" (what we refer to as hidden variable theories in this context). They are "nonlocal" because measurements on a system can instantaneously affect entangled systems regardless of their spatial (or temporal, for that matter) distance. However, crucially, these nonlocal correlations are not at the level of measurement outcomes. There is no way to measure a system $B$ and be able to infer anything at all about what happened to another system $A$ outside of its light cone. There is no way to observe any effect on $B$ "caused" by anything outside of its light cone. This is also sometimes referred to as the no-communication theorem. See also Does Bell's theorem imply a causal connection between the measurement outcomes?.
Let me try and be even more precise about this: these nonlocal correlations I'm referring to are observable (otherwise they wouldn't be a significant part of the theory at all). But you cannot observe them performing any individual experiment. Meaning regardless of the shared state $A$ and $B$ have, whatever measurement they're doing, they won't be able to infer (superluminally) anything about the way the other party interacted with their state. However, if $A$ and $B$ repeat their experiment multiple times, changing the way they interact with their system each time, and then at the end go and compare all the measurement statistics they found, then they'll observe that the observed correlations are "nonlocal", meaning that the state they shared was "more correlated than classically possible".
Note that the above makes no reference to whether the quantum framework we're using is relativistic or not. In fact, the standard nonrelativistic formalism for quantum mechanics does allow superluminal communication, simply because special relativity is not taken into account at all in the formalism. But this is completely different from the issue of the nonlocal correlations due to quantum entanglement/correlations.
You can fix the relativistic issues by integrating relativity into the formalism, as you do in QFT, and then you get back the expected result that quantum mechanics is local (again, in the relativistic sense; the features about "nonlocal unobservable correlations" described in the first paragraph remain).
A: This is a bit of a misunderstanding, Einstein strongly disliked the probabilistic nature of QM, and has a few quotes speaking out against the budding feild. One of the more famous ones being "God does not play dice"
At the time, locality was well accepted, so rather than being supportive of his colleagues findings, he used the term "spooky action at a distance" as a dismissive and damageing jab making fun of QM. And although Einstein eventually accepted that quantum mechanics was an experimentally confirmed theory, he never excepted some of its features.
