If you follow the history of currency from a piece of gold of unknown purity, to gold coins produced by governments, to the establishment of central banks, to outlawing privately issued banknotes, to fiat currencies...
...you'll notice that all of these are created to keep up with the demands of a technologically driven advancement in the state of economic affairs.
If I showed up to your store with a lump of gold for trade, we would have to go find someone we both trusted to figure out how pure it was. That takes time and money. That inhibits commerce from taking place.
Having the government get involved and eventually come to manage the digital exchange of currency would be no different than these analogous historical transitions.
It doesn't make sense to completely rely on private companies for a type of economic transaction that has already become the dominant way that value is exchanged. Especially when those private companies require the full backing of nation-states and their currencies to even have any sort of business model to begin with. Why else would Visa and MasterCard have not yet created their own currencies?
You can't have a free market if you don't have a viable marketplace to begin with.