They don't need to mint a coin, that's just theatre. The US treasury could simply create an amount of dollars equal to the debt and pay it off right now. The country would instantly have no debt. The UK could do this too, as could any country and we would not need any austerity measures.
The reason they don't is that the main creditors i.e. the Federal Reserve which is a privately owned bank run for profit, and the other large banks would no longer receive hundreds of billions of $ in interest each year from our tax money. Currently our taxes go to the government who pay much of it as interest on the government debt so it end up in the pockets of bankers and families that own the banks (tax redistributing from the poor to the rich). The money that the banks lent the government was created out of thin air via the fractional reserve system. The fact that very few politicians in the US or UK admit the debt could be paid off immediately at any time easily speaks volumes - they clearly aren't working for us.
In fact all the money in the economy is created as debt, the money supply expanding each day by the amount of new loans that were taken out that day. The banks simply create the loan money out of thin air in their computer systems and loan it at interest. The money supply is simply the aggregate of all the current loans. But the money for the interest that needs to be repaid is never created by the banks, only the principle. Thus it is impossible for the debts to ever be repaid, and a proportion of people have to go brankrupt each year and more debts need to be taken out to repay the interest on the current ones. This explains how nearly all people, all governments and all businesses can be in debt at the same time without a counterpart group of people, governments and businesses who are in so much credit that they could afford to lend it to the rest.
The truth is all our money has been born of debt through bank loans. Any government treasury can create its own money, after all if the government grants the privillege of creating money out of thin air to banks, it can grant it to its own treasury. Lincoln did it with his greenbacks - notes that ere created by the treasury as a credits, with no interest to pay.
And don't be fooled into thinking that we would get a huge wave of inflation if the treasury creatd its own money - it would be no different from allowing the banks to create it except interest would not be charged so our taxes could be massively cut. People forget - income tax was a flat 6% 100 years ago!
In Britain, there is one fast growing group who is managing to get the message out and a few honest MPs have got involved:
http://www.positivemoney.org/