I am having trouble wrapping my head around how it's possible that there are no local hidden variables in quantum mechanics. I came up with this argument that seems to me to be solid, but it would contradict the nonexistence of local hidden variables. Could someone point out where the argument goes wrong?

Suppose Alice and Bob take two entangled particles very far from each other. For simplicity, assume they are 1 ly apart, and with a relative velocity of zero. Suppose the particles become entangled at year 0.0, and Alice and Bob take 5.0 years to reach the 1 ly mark.
When I talk about "before" and "after" in this argument, I am referring to the reference frame of a point exactly in between Alice and Bob, also with a relative velocity of zero.
Suppose that the particles are entangled in the Bell state, so that A's and B's measurements will always produce the same binary result of either 1 or -1.
Without loss of generality, let's say Alice gets a result of 1. Now, Bob is guaranteed to measure 1 as well. This is true matter how long he waits. He could wait until year 50.0 and he would still get a result of 1. Consider two scenarios:
(1) Alice mesaures at year 10.0, and Bob measures at year 10.5
(2) Alice mesaures at year 10.0, and Bob measures at year 9.5
From Alice's perspective, there is absolutely nothing different between the two scenarios. In either case the no-communication theorem prevents any information from being transmitted, since they measure only 0.5 years apart. Thus, she will measure 1 in either scenario.
Now, consider a third scenario:
(3) Alice mesaures at year 9.0, and Bob measures at year 9.5
From Bob's perspective, this scenario is identical to the second scenario (due to the no-communication theorem), so he (and by extension Alice) is still guaranteed to measure a 1 in this case.
Now, consider a fourth scenario:
(4) Alice mesaures at year 9.0, and Bob measures at year 8.5
Again, from Alice's perspective, this is completely identical to the third scenario, so she (and by extension Bob) will end up measuring 1.
You can probably see where I'm going with this. We can follow this logic all the way back to the creation of the entangled pair. The conclusion of the argument is that if Alice measures a 1, then it must have been the case that measuring the particles at any previous point (after they were entangled) would also have resulted in a 1.
This seems to me to force the existence of a local hidden variable. Apparently this isn't right, because we have recently ruled out local hidden variables from being consistent with observations, but I am at a loss to see what is wrong with this argument. Or is what I said correct, but there is another explanation for it without involving a local hidden variable?
Could someone explain what is the issue with this argument? And, could someone explain Bell's theorem/Bell's inequality in layman's terms, and how it's possible that QM can violate it?