Bell's Theorem hasn't been raised here. It should be. Either Einstein is wrong or Bell is.
I have puzzled over this for years, decades. I think Bell is wrong but I haven't been able to design a proof (Or I'd get the Nobel prize!)
Could Physicist or one of you other qualified physics types take a wack at this? I don't expect a definitive answer, I don't thing there is one.
But is there as big a contradiction here as I think there is?
I can't handle this one. Perhaps one of the others can. I don't have a separate list of physicists, but those I'm pinging will know who I've missed.
What you probably mean is the apparent contradiction between Special Relativity and Einstein-Podolsky-Rosen correlations. The fact of the matter is that there is no contradiction. EPR phenomena are correlations, not causal relationships. To borrow a clause from statistics, "correlation is not causality".
Quantum mechanics predicts EPR correlations. It predicts them without postulating any hidden variables, and without postulating any sort of signal between one measurement and another. Such a signal might violate SR--QM is agnostic about the structure of spacetime--but the fact that QM demands such correlations whether such a superluminal signal exists or not should tell you that it's a philosophically unnecessary concept.
[Geek alert: it is possible to construct interpretations of QM where a superluminal signal exists (e.g. the Bohm interpretation), but it's a very peculiar kind of signal, and it cannot be used to send information.]
EPR correlations cannot be used to transmit information faster than light. In the case of the Aspect experiment, the correlation between the two polarization states cannot be seen until the two answers are brought together (a process that is slower than light), and compared.
What the correlation represents is the limits of information. The reason why the two measurements get the correlated answers is because they're extracting the same piece of information...and this despite the fact that the information wasn't in a definite state until it was measured. (If it were in a definite state, it would have to obey Bell's Inequality; this is what sank Einstein's interpretation well after his death.)
"A-ha!" you say. "A-ha! I have you. You said 'until'! What does 'until' mean, when the events have a space-like separation, and the second event isn't causally allowed to know the outcome of the first event? How can it know when to collapse, without a superluminal signal?" Good question, well put. It's tempting to say that since measurement A happened first, it therefore caused the outcome of measurement B to turn out as it did, and because they have a space-like separation, there must have been a superluminal signal. I can turn this around, however: because the events have a space-like separation, it's always possible to choose an inertial frame where measurement B occurs before measurement A. A similarly prejudiced observer would postulate a superluminal signal going from B to A. The information can't flow in both directions, so it's hard to call it a signal, exactly.
So there you have it. The correlation is required to exist. You can postulate a signal to do it, if it makes it fit more comfortably in your imagination, but the signal needs to work equally well backwards, too, which is an uncomfortably weird property to imagine, and mathematically you don't actually need it, anyway.
There are other interpretations, besides. I'm leaning towards the "consistent histories" interpretation, myself. (The measurements occur more or less independently in different, parallel realities, but the realities in which the measurements don't properly correlate get cancelled out, and so are never seen. It makes things like the Elitzur-Vaidman bomb testing problem easier to understand.)