I think David Rabahy's comments illustrate a problem with OP's postulated scenario.
Individuals don't want to hold something that is fixed in value. They would rather hold something that appreciates in value. And markets are merely collections of individuals. As such, market forces will not conspire to make all monies asymptotically approach stable value, as long a there is an alternative available that appreciates in value. If indeed market forces will drive all monies to approach some prototype. The market forces in such a scenario would conspire to make all monies approach that other value-increasing alternative.
1) I am not wrong, and its Nash's postulate. 2) You have contradicted yourself by admitting the truth of this. 3) The reason bitcoin has value as a gold is because our money systems are instable DUCY this is relevant?
The sub title of ideal money is "the motivations of savings and thrift" and the motivations of savings, or want for an inflation hedge, is inflating fiat. Bitcoin is valuable when fiat sux.
So it occurs to me to think that that which is not achieved by a grand action of establishment by fiat may alternatively tend to come into existence as a consequence of a process of evolution. And of course, after a certain degree of progress by evolution the rest of the progress could possibly be realized by a convention or a process of fiat.