They will eventually. A new thing will come out to peg a coin against and all currency will peg them self to that thing. Token or coin. It would take an investment of about 2.5 million to establish a stable coin token for other Crypto and FIAT currency to peg themselves against.
The value of this coin would not change. what each other currency pays for it can vary, however the price of the token never changes. This is very possible to do ;)
This is not a token that would be worth to invest in, unless you were investing to get the token established then it would offer returns. It would not offer much benefit to be traded on markets though. For pension funds and savings it might work a bit better as a security that you will always have that amount.
There are one or two small issues that would need to be resolved but truly they are minor issues. If I cannot get funding to do this. Someone else will, It is only a matter of time before the mass catch onto common sense. Although more often it seems things don't go that way. Someone with money will or we are all doomed to eternal financial slavery.
A central coin token will be established that even the FIAT will peg their exchange rate against and depending on how well a country does in comparison to others, will determine how much of that currency a token or coin will cost to purchase.
I think that the achieving of this independence is almost impossible because Bitcoin for Cryptocurrencies is like USD for fiat currencies. Also all assets can only be traded against Bitcoin and they depend directly on the Bitcoin price depending on the situation.
In addition there is a one-sided dependence between the majority of Blockchain projects and its alternative digital currencies and bitcoin crypto. Smaller currencies or start-up projects are particularly exposed to the partly enormous effects of Bitcoin's performance. This can be attributed to the direct trade between Bitcoin and these currencies.