Isn't this more or less what the Aspect experiments did?
No.
This is the conceptual crux of the issue that I am talking about. The various Bell experiments establish violation of local realism. They do not establish that the collapse of the state vectors are events which are spacelike separated; if they did, the No Communication Theorem would be violated.
We can assume that they are. But we cannot prove that they are, because we cannot transmit the results of our measurements to each other in any Lorentz frame faster than the speed of light, and until we communicate our results to each other, neither of us actually has experimental evidence of what the other already knows.
And notice also how stringent this condition is: It holds in every Lorentz frame. If it did not, we could put a "man in the middle" who could violate causality.