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.
Shannon's model, for instance, does not require bits to be binary or digital. The model has been quite effective in molecular biology (cancer research and pharmaceuticals) where the message is in Genetic Code and the sender/receiver are molecular machinery.
Information in Shannon's model is the reduction of uncertainty in the receiver (or molecular machine) as it moves from a before state to an after state. The model entails these specific elements: message, sender, encoding, channel, noise, receiver, decoding.
I cannot accept the use of term "information" in an application where all these elements do not exist, e.g. when it means no more than determinism or physical causation.
Again, information is the successful communication of a message not the message itself. It doesn't matter what the message "is."
EPR is strictly deterministic. Information cannot occur via EPR simply because the sender cannot encode a message by measurement (the No Cloning Theorem.) The sender has no control over the outcome of the measurement to encode a message for the receiver, Lorenz frames (No Communication Theorem) notwithstanding!
By the way, Rosen extended Shannon's model for biological systems to a circular model. It's very promising. If you're interested, pick up a copy of "Life Itself."