And yes, I do mean five dimensional physics as the most promising explanation for phenomena such as EPR (in my view.) For more, please see P.S. Wesson's Quantum Mechanical Consequences of Five-Dimensional Relativity
Roger Penrose also observed that a simultaneous measurement of a spatially separated entangled photon, would result in an unacceptable paradox where the measurement of each determines the other based on our current physics. He calls for a new kind of physics, but again I suspect the solution rests with a higher dimensional dynamic rather than a 4D or compactified extra dimensions (Kaluza/Klein.)
Your objection is noted, and wrong. If we accept it, we have to accept a definition of "information" which has no capacity to model physical reality, which would be ironic given Shannon's original motivation, to say nothing of bizarre.
Spin flipping is the closest thing in the physical world to the transmission of bits, which is Shannon's basic definition. There is one possible symbol, and the channel is loss-less.
If an eigenvalue of a state vector could be deterministically measured locally based upon the collapse of an entangled state in a remote experiment, the remote measurement would indeed transmit 1-bit. That one bit of information could be used to effect -- there could be physical outcomes chosen by the experimenters. For example: if the bit flips, fire a laser which travels back to the source, killing the remote experimenter [or his cat.]
There is no difference in terms of the information transmitted in this Gedankenexperiment with one in which the experimenter transmits a laser pules which is pre-agreed to mean "kill me." [Are you of the opinion that Paul Revere's friend was not transmitting any information about the British by essentially using the same method?]
There is one important physical difference, even if there is not a mathematical one.
In the EPR version, the influencing events are not within each other's light cones. Consequently, there would be Lorentz frames in which some observers observed the measurement before it occurred, and observed the remote experimenter being killed by the laser before the spin flip occurred. That is the reason for the No Communication Theorem. Because we understand that if communication was possible via this method, information would be exchanged. It isn't, and it doesn't.
I find it interesting that Wesson, in this recent research, is broaching the idea of a "cosmological constant."
Albert Einstein, whether by instinct or mathematical intuition or whatever, felt pretty sure that a cosmological constant existed. He pursued the quest for it. He did not find it. Late in life, he allowed as how the search for the cosmological constant was the biggest mistake he made in his scientific career.
And it seems the next generation of physicists dropped the matter accordingly.
Then again, just because Einstein couldn't find it, doesn't mean that it isn't real, that it isn't "there."
Maybe P. S. Wesson will have better luck!
If found, it could help to elucidate many currently intractable problems in theoretical physics.... In particular, the reconciliation of the general relativity and quantum theories.