Proteins do not contain messages. Amino acids do not contain messages - hence the Urey/Miller experiments made no further progress whereas the Wimmer experiment did.
If one is focusing on the molecular biology, the observation that prions have no message is moot.
Both in the above article and in our book, Timothy - the spotlight is on "information theory and molecular biology" as an integrated whole.
Information is simply a dead end if all the “information” in DNA tells you is how to make functional RNA transcripts, and all the RNA transcript do is control and regulate and code for protein production.
Does this compute to you?
DNA codes for functional proteins. The only “information” in DNA is how when and where and the recipe for a functional protein.
If the protein doesn't contain any message or information then the “information” is a dead end.
Proteins convey information in chain reactions. Check out “Signal transduction” involving (usually) the phsophorylation and inactivation/activation of proteins by phosphorylation information chain reactions.
For example....
A small molecule binds to a cell surface receptor protein. That protein then is activated and posphorylates a protein that holds inactive a transcription factor protein, and that phosphorylation makes it let go the transcription factor. The transcription factor is then free to go into the nucleus and bind to a specific DNA sequence and turn on a specific subset of genes to produce a specific subset of proteins necessary to respond to whatever that original small molecule was telling the cell to do.
If “information” means anything than it is obvious that proteins contain and convey information.