Banks that offer depositors gold redeemable money would still issue fiat money.
Unless this term has taken on a credible meaning in jargon of which I have somehow remained unaware, I feel the need to make a point about usage of "fiat".
As commonly used, "fiat money" and "fiat currency" is either conflated with or taken to imply "fake" or "unbacked" and therefore valueless. This is mistaken. The meaning of "fiat" is "mandated" or "decreed". All government-mandated monies and/or currencies are fiat by definition, even when backed by and redeemable in gold. In a free money market there is no such thing as fiat money or currency, no matter what its structural nature. It is precisely because the markets are free that nobody holds authority to decree one money to be accepted and all others impermissible.
Free and competing monies can bear any structure you care to name. They can be pure paper, backed in gold, corn husks, beads, dried chicken feet, and so forth, redeemable or not. The key to successful money under "normal" market conditions is faith. If there is no faith in a currency it can be solid gold of absolute purity and yet it will be useless, and when there is need the most seemingly mundane commodity may purchase a kingdom. There are the old stories we used to hear when I was young about cities such as Dresden and Berlin after capitulation where women would rent their vaginas to soldiers for a cigarette or two. While possibly and even probably apocryphal, those stories nevertheless illustrate the point about what constitutes "money": that which facilitates trade under a given circumstance.
The OP was right in that given the right circumstances gold is worthless. When an economy collapses, and I mean REALLY collapses, gold is likely to become worthless because one will be concerned with his ability to eat on a given day and gold, after all, holds precious small nutritional value no matter how much of it you stuff into your mouth.
It seems to me that many people make good points about money and gold, but they fail to see the larger picture, thereby disabling them from pulling the disparate good bits and pieces together into a coherent and sensible gestalt. I am not sure even I am able to do it, but allow me to make a few observations.
Firstly, I will state unequivocally that gold, silver, and other PMs are as good a basis for money as any other and better than many.
Next, the value of any given money system lies not nearly as much in its structural composition (gold, toilet paper, what have you) as in its stewardship. However, I suspect that convertibility should be an absolute requirement. This, however, depends greatly upon the precise definition of "convertibility", which I will deal with shortly.
Next, any system of currency that can lay claim to being actual "money" must by definition be devoid of the interest component. Otherwise you have no money of which to speak, but only debt. Yes, even solid gold can become pure debt when mistreated in such ways. Another way of expressing these two ideas is asset-backed currency v. debt-backed. The former is money, whereas the latter is just debt.
Next, gold is superior to most other standards precisely because it is: a/ gold, and b/ therefore, perfectly uniform as a standard and thereby virtually immune to value manipulations. There have been suggestions of "commodity backed" currencies where the "money" in question is backed by a so-called "shopping cart" (a la the consumer price index [CPI]) of commodities. This system, however, would be grossly inferior to that of a gold standard and here is why: gold is a simple commodity which in turn would constitute a simple standard that introduces no complex interactions between the relative values of the commodities comprising the standard. Those relative-value relationships, which arise due to the multivariate structure of the money itself, in turn give rise to complexity in the standard, particularly in determining those relative values. It is precisely this complexity that provides the jungle-matrix wherein manipulation may be well hidden. When factors are complex and subtle, as they most likely would be in a commodities-backed monetary system, the foxes will be at play doing what they do best, stealing your chickens out from under your nose while you're busy doing other things.
Where money is concerned, structural simplicity is the grandest virtue of them all, for it provides the greatest transparency and the best guarantee against the sorts of manipulations against which the OP complains. Gold is effectively eternal. It cannot be manipulated without being detectable. Where monetary standards are concerned, eternalness is next to godliness. Couple that with an absence of the interest element and the markets become far more predictable, stable, and therefore prosperous. Barring significant disasters, markets become rock-solid and economies thrive as a result. Because there is no built-in inflation due to the debt component of the currency, the only factors driving price are supply and demand. The world becomes a far simpler and more predictable place in so many ways because the bankers and money speculators will have been de-fanged. There would be no place in the world for such people and we would all be far and away better off.
Finally, on the point of "convertibility", all monies are convertible. I can take an FRN, go to the store and trade it for, say, a tube of tooth paste. I have converted my "money" into tooth paste. The problem is NOT convertibility per se, but rather the stability of value of that piece of paper that I hand the cashier. The interest-component of a debt-based currency is what guarantees the instability in price no matter what the physical commodity conditions of the tooth paste industry might be. Compare that with an asset-based currency (actual money) where changes in aggregate supply and demand are the only major factors affecting price, making such currencies far less prone to speculation and other manipulations.
One sees readily just how much room there is for improvement upon the system we currently "enjoy".