Why is it that Bitcoin be used to fulfill this role (or the role of a best-effort approximation of Nash ideal money)?
Simple 'yes' almost lost in superfluous verbosity duly noted. I think I may be on the verge of understanding your argument, though.
Yet you say that gold no longer has the gold like properties you seek. Stop saying 'gold' when you really mean 'as stable in value as humanity can muster'.
(apologies for the edit - if nothing else, it can serve to inform you of how I am reading your statement)
I can see that a Nash ideal money is a very important thing. If we had something that acted like Nash ideal money, I agree that it would be an important advance for humanity. I am skeptical of your seeming insistence that Bitcoin be the thing that fulfills this role. Indeed, it has several strikes against it already:
1) Its value has been increasing at an astonishing rate over its short history
2) Almost every stakeholder already wants to increase its utility, and thereby consequently its value
A) I don't think it can be decreed to make the value of Bitcoin stable
B) I think it likely inevitable that it will eventually serve as coffee money.
To clarify A) - the paltry fraction of humanity that is current stakeholders of Bitcoin have a natural incentive to increase the price of Bitcoin. Call it a tragedy of the commons if you wish, but human nature will conspire mightily against any attempt to freeze the value of Bitcoin as a whole. No matter how great a tool Nash ideal money would be for humanity as a whole.
I can see that increasing the tps will add value to Bitcoin. I can see that this impacts negatively on Bitcoin's ability to serve as Nash ideal money. But it already fails spectacularly at value stability. I think you need to look elsewhere for a candidate for Nash ideal money.