Determining interest rates and determining the nominal value of currency are two sides of the same coin, so to speak. In order to be useful, the nominal value of a currency should be relatively stable. As you've noted, due to nominal rigidity, it is also beneficial if the value of the currency depreciates over time. If you take these two conditions as a requirement, you arrive at average inflation targeting (which is what the US Federal Reserve currently does). As a final condition, money needs to be widely used in order to be useful (which is why you cannot simply set up your own Federal Reserve).
Could the task of determining the supply of money be taken over by an institution other than the Federal Reserve? If you try and do this in the real world, you will likely get a lot of push back from the actual Federal Reserve. But if we ignore the switching cost, there's no reason that we couldn't have the same job done by a decentralized institution (say perhaps on a blockchain).
What would be the benefits of this? First, it would eliminate the fear that the Central Bank my change rates due to political pressure. Second, it would make it harder to monitor and censor unpopular financial transactions. Although depending on your point of view, this might be a bad thing.
How will this play out in practice? I suspect that countries with poor central bank independence will gradually lose the ability to issue currency as citizens move to safer alternatives. More importantly, better financial tools will allow people to hold their savings in assets other than money and quickly change between types of assets. I expect some sort of "robot advisor" will eventually take over this job for most people.
Of course, this is assuming that the singularity and post-scarcity don't make the whole concept of money irrelevant for 99% of people.
I'm curious what people think is most likely say 50 years from now?
Low interest rates are like giving free money to banks etc . Why not just give us ordinary people free money ? Why don't I get to borrow at the fed rate ? I will happily provide colateral. The right question is not why the fed sets interest rates, but why it does not provide the same rates directly to everyone including us little folk while cutting out the banking middlemen. We should all have a a fed account.
One common free-market argument against government control over interest rates goes something like this:
In other words, they are arguing for some variant of free banking.
I find this to be a thorny issue to think clearly about. Theoretically, the free-market argument makes sense. But, on the other hand, we have nominal rigidity and the risks involved with change. Also, some governments are more competent than others — the right answer for Switzerland may be different than the right answer for Zimbabwe.
Note that I think the political feasibility is very low. Government control over the money supply seems to be a stable equilibrium. So this question is mostly of academic interest. But it might be relevant for, e.g., charter cities.