It should be completely uncontroversial to say that US treasury securities (the constituent parts of the national debt) carry no default risk, because the government that issues the securities is also the issuer of US dollars. That is, US debt can always be redeemed with new cash. This option gets pejoratively called a "soft default," or "inflating away the debt," but let's think that through the rest of the way.
Treasury securities can be held by banks. They can also be held by private investors, both domestic and foreign, though they aren't as important to this argument.
Banks hold treasuries for the reason indicated in the linked graph. To clarify that, the blue line is the fed funds rate, which is what people are referring to when they say "the interest rate," and red and green are the yield of three-month treasuries. The fed funds rate is the benchmark cost of lending, so it is controlled by the Federal Reserve in its attempts to manage the money supply (since lending creates money).
In the graph, the fed funds rate is basically indistinguishable from the T-bill yield. The reason for this is simple: The fed funds rate is controlled by setting the opportunity cost of cash (the opportunity comes from interbank lending). And cash and T-bills are both risk-free assets, so when you include the discounting for holding a bond versus cash, they should have the same opportunity cost.
A consequence of these facts is that, if the T-bill yield were for whatever reason higher than the Fed's desired fed funds rate, banks would buy them because they'd have higher yield than the bank's cash, which would make reserves scarcer, which would drive up the fed funds rate, which would force the Fed to intervene by -- lowering the T-bill yield by buying them up.
Note that there's no role in this story for "bond vigilantes," and the reason is that bond vigilantes can't exist. The treasury yield is always going to revolve around the fed funds rate, and no bond vigilante could hope to stand against the central bank. So the point isn't even that the public debt should be abolished (though it could be done, if one had the willpower), but rather, that the yield on treasuries is controlled by the Fed, not by the bond markets.
Investors who don't fancy themselves vigilantes, of course, want treasuries in their portfolios as risk-free income, but they too must bend to the will of the Fed, because the Fed always has more money than them.
There's another somewhat subtle point regarding banks that I may as well bring up. When the government sells a bond to a bank, it reduces bank reserves but does not reduce deposits (i.e., MB goes down, but M3 does not). But the government doesn't sell bonds for kicks; it does so because it has a spending mandate, so that money shortly returns to the economy. This returns reserves to their previous level, and increases deposits. In this sense, banks have an effectively unlimited demand for bonds, since when it comes to excess reserves, some yield is always better than no yield, and the money spent on bonds is always replenished by subsequent public spending, so there is no "market test" for public debt.
Remember also that inflation only came up because of the fact that bonds can be redeemed for new cash, but, outside the central bank's liquidity management operations, the fact that they can be means they don't have to be: the possibility is enough to render them risk-free. The greater question of inflation is more complicated, though it's certainly not the bleak picture it's often made out to be.
As another side point, since there's an inherent tendency to say that public debt is bad, something else to keep in mind: Cash is a liability on the Fed, and for the Fed to have positive equity, they need either a balancing negative liability (i.e., the Treasury could run a persistent negative account at the Fed), or they need assets corresponding to their liabilities (i.e., assets need to back all the base money in circulation), and for the most part those assets are -- treasuries. As a simple matter of accounting, the only way to lower the public debt is to also destroy money in equal quantity. So another term for "public debt" is "public liability side of nongovernment financial assets."
Now you might be wondering, why was there an actual, existent public debt crisis in Europe? The answer is simply that this anecdote does break down in Europe, and it does so right here:
- That is, US debt can always be redeemed with new cash.
No single nation has the authority to issue euros, so whether their bonds default, in the final case, is determined by the politics of the supranational European Central Bank. This meant that treasury bills in Europe contained both discounting and a risk premium, and the latter did not have to be controlled by the ECB as a matter of monetary policy. The EMU's debt crisis went to the back burner when the ECB actually did decide that it would stop sovereign bonds from defaulting, and the real problem became replaced by the completely imaginary problem of government deficits being "too large," with no clear meaning of the word "too."