Many people depend on Treasuries for income. People depend on the money from interest or principal payments of treasuries to show up in their accounts so that they can spend money as planned. Treasuries are perfect for tight tolerance, high dependency financial situations. Money is disbursed via the FedWire system which, in my experience, can put money in your account within minutes. That the Treasury United States of America will pay its debts on time has been a reliable fact for a long time now.
For many holders of Treasuries, a little delay in getting their money wouldn't be such a big deal. Getting paid on Tuesday instead of Monday or the previous Friday wouldn't affect my cash flow terribly. However, there are holders who depend on timing of payments much more than I do.
One of the largest uses of Treasuries in the financial world is called the RePo market - overnight lending. Banks let each other borrow Treasuries for a very short term (often overnight) in exchange for cash. Since Treasuries are a sure thing, nobody has anything to worry about. Treasuries are perfect loan collateral.
But the United States is dangerously close to defaulting on its payments. If this occurs, it will more likely be an interruption to the market than a 'real' default. There are, however, billions of dollars of payments riding on the timely payment of Treasury debt. On top of that, probably trillions of dollars of derivatives are priced based on Treasury rates. Large banks depend on huge sums of money changing hands all of the time, this interruption will be an immediate credit crisis for banks.
There may be some options for Treasury holders, and some may be able to cope with a delay in payments, but there will be some holders who get put in an extremely tight spot. Whether they have real cash needs that were meant to be funded with that, and cannot pay bills (including their own debts) or have a line of credit that's interest payment spikes due to the crisis, they will be pinched between a Treasury unable to pay them and customers demanding payment.
It is annoying to have your paycheck delayed by a day or so, it is disastrous to have your paycheck delayed when you have a major purchase to make on that day.