The value of the dollar derives from trust? Trust in what, how? That is an incomplete statement. It is unipolar and makes no sense, it is yin without yang, plus without minus. Trust does not stand alone, it requires a partner, it must be trust in another to do something or not to do something, to fulfill a promise. I suggest that the value of the dollar back then derived from the trust that precious metal coin was backing the paper at a ratio of 1 to 1, and that redemption on demand would be honored. In other words, they trusted that they could get their gold. Therefore the value of the dollar paper derived from the gold. People knew they were trading title warehouse receipts for gold and or silver. People often ran on the bank to get out the metals when they felt the government was not backing sufficiently, the system was self balancing. If you had told one of them that the value of the dollar derives not from the gold, they would have laughed. Even our dollar today, trust does not enter into the equation, we accept dollars because we believe and hope someone else will accept them when we go to spend them, and they are receivable for taxes. Often we don't even question that others will accept them; but we understand and expect others to raise their prices which is in a way like conditionally not accepting them...at the old price, that when they lose their value to inflation, the merchants say I won't accept them at the old price but I'll accept them at the new price, which means you pay more.
You need to understand the history of paper money in this country. It started out with banks issuing notes which they were obligated to honor. They got passed around like money for years, as though they were worth something because of the backing by a bank, but in many cases, they were not, either because the bank was out of business, or because it did not have the resources to honor the notes anyway. Eventually, this practice evolved into federally issued notes, and then into paper money.
It gets passed around as though it has value, but the value does not spring from some backing. There has been no backing for paper money since the Great Depression, yet, it continues to have value. The value really springs from trust--trust in a lot of things: Trust that the government is not going to vastly increase the money supply faster than the economy grows. Trust that the paper money will continue to be "legal tender" for debts. Trust in the economy. That's why I just said "trust." You can't specifically identify what it is that we are trusting in. There are lots of things.
You are right that it used to be that it was trust that someone would back it with something else like gold or silver, or anything you like. But those days are gone, and have been gone for a long time. They are gone for the simple reason that people eventually realized that very little of the paper money ever actually got turned in for gold or silver, and that the real utility was not that it represented a right to payment by the issuer, but rather that it was accepted as a medium of exchange by people with whom you are dealing directly.
That acceptance, however, depends on trust. If I thought that I would not be able to use money I receive for payment of my own debts, I would not be willing to accept it. But because I trust that I will, I am willing to accept it.