Assuming a gold standard were possible (i.e. that people wouldn't just create fiat money around it), it doesn't seem possible that we could get optimal economic growth under a gold standard.
In market/capitalist economics, Y * P = M * V: Real gross domestic product (Y, the total production of final goods and services) times the price level (P) equals the quantity of money (M) times the velocity of money (V, the number of times, on average, a unit of money is used in a transaction for final goods and services). We can also say Y = (M * V) / P. We can choose our units so that P = 1; therefore Y = M * V.
Remember that V in this case is not just the number of transactions using a particular unit of money, it is the number of times the unit of money crosses the production/consumption boundary. Alice's employer pays her; that's one transaction. Alice buys food at the store; that's two transactions. The store uses the money to buy food from a wholesaler. That's not a transaction that counts towards V, because the store and the wholesaler are both on the production side. Similarly, when the wholesaler pays the processor, or the processor pays the farmer, or when the farmer pays his mortgage, those are not transactions that count in the velocity of money. It's only when the store, wholesaler, processor, farmer, or banker, pays an employee, investor, or landlord does the money cross the producer/consumer boundary and V increases. It seems clear, therefore, that V is physically limited. No matter how an ounce of gold is represented, whether it's a physical coin, an ingot, or an entry in a database that corresponds to some physical gold, money can move only so quickly.
The point of a gold standard is to physically limit M, to create a quantity of money whose increase is physically limited. Since V is already physically limited, that means the quantity of money in motion, M * V, is physically limited. If we have price stability, then P is constant. Therefore, an increase in real gross domestic product is limited to the limit of the increase in the physical quantity of gold. But there is no particular reason to believe that the limit on the increase on the physical quantity of gold is the optimal increase in real gross domestic product. And that's absolute GDP; there's no reason to believe that the increased quantity of gold even match even population growth.
Furthermore, what is the value of a specific quantity of gold? Gold has very little intrinsic value, outside of electronics and jewelry: you can't eat or drink it, nor does it keep you warm or dry. Gold does, however, have a physical cost: the marginal socially necessary abstract labor time to produce the "last" unit quantity of gold. Expressed in money, the marginal cost of one unit quantity of gold is one unit quantity of gold. In other words, all the economic growth would go to the gold producers. There really isn't a world with an industrial economy in which a gold standard is a good idea.
There are, of course, any number of people that support a gold standard who have just not thought through the economics. But there must be gold standard advocates who have thought through the economics. Some of them, I suppose, really would limit real economic growth, but I've never seen a gold standard advocate say explicitly endorse limiting economic growth to the increase in the supply of gold. The others, however, realize that some sort of fiat currency is necessary for economic growth in an industrial economy. The question, then, is who gets to control economic growth, and, more importantly, who gets to distribute the benefits of economic growth. It is fairly obvious, then, that such control would go to the people that own most of the gold.
Economic power, control over the means of production, is political power. One does not imply the other; they are the same thing. In an advanced industrial economy, the means of production is not ownership of factories or land, but ownership of money itself. By definition, the government is the set of institutions and their members that own the money. Thus there can be no such thing as "private" ownership of money: the fundamental owners of money are, by definition, the government. So the question is not whether we should have government ownership of money, but rather what kinds of institutions, with what kinds of members, should constitute the governmental ownership of money.
I'm not a big fan of republican democracy, but a republic is, at least to some extent, accountable to all the people. Putting the government in the hands of people who say proudly and explicitly that they recognize no accountability except to their own well-being seems like a Very Bad Idea.