How does government control of the means of exchange, per se, affect workings of the market?

It completely destroys the possibility of a free market. When the value of the means of exchange can be unilaterally and arbitrarily altered by a central power there is no price discovery. This would be true even if we were only mandated to use a single government issued currency, and the currency itself wasn't massively manipulated by the FED (it is).

In a free market, people would be free to choose what sort of units of exchange they wanted to use. They wouldn't be forced to use government fiat that is inflated away on the whims of the oligarchs on Wall Street. Units of exchange that were responsibly managed and credible would gain popularity, while volatile and unreliable means of exchange would be abandoned. Dominant units of exchange would start more accurately representing the value of what those units were worth in terms of what they could be used to purchase or barter for, rather than how many trillions are printed by a central bank.

In a free market, people would be free to open banks and accept deposits and issue loans in any of these units of exchange that drew demand. They wouldn't be forced at the point of a gun to participate in a government mandated ponzi-scheme in order to make the basic purchases, trades and payments that life requires.

Whether people are exchanging for Bitcoin or pukka shells, the principals of market economics would seem to apply nonetheless.

Not only does the government prevent you from exchanging different forms of exchange (such as bitcoin) through absurd regulations, taxes, and fees, but they demand that those taxes, fees and fines are paid in government fiat dollars - maintaining their monopoly.

We do this not only in our own country, but on a global scale, through the use of the US dollar as a "reserve currency". Forcing other countries to use our country similarly destroys markets and creates a huge artificial demand for US dollars.

https://en.wikipedia.org/wiki/Reserve_currency

Internationally, like domestically, we enforce our market manipulation at the point of a gun. What happens when a country wants to go off the US dollar as their reserve currency? You could try asking Saddam Hussein and Gadaffi.