With a fabulous magical gold energy technology, and gold hard to extract, It's clear that a gold engine would have to be fantastically efficient, and that is bad news for gold.
Start by comparing your gold engine to a gasoline IC engine. Yeah, yeah, diesel is more economical, but not that much more. Gasoline has an energy content of about 46 MJ/kg. Assuming that a gas engine has a thermodynamic efficiency of 30%, that's an effective energy content of 13.8 MJ/kg, or 37.6 MJ/US gal (you can tell I'm American, right?) With current automobile gas efficiencies in the neighborhood of 1 gal/hr, assuming 30 mpg at 30 mph, that suggests a nominal automobile power of 10,444 J/sec, or about an average of 14 hp, which is not clearly wrong by an order of magnitude. At 2 dollars/gallon, this gives an energy price for gasoline of about (roughly) 5 cents per MJ. With gold prices currently at about 1200/oz (troy), it would take about 50 mg of gold to buy a gallon of gasoline.
In 2014, world production of oil was 86.8 million bpd, or 31.7 trillion barrels. A petroleum barrel contains 31.5 gallons, so world production was (assuming all petroleum is gasoline) of about 1 quadrillion gallons. At current gold prices, that's about 50 trillion grams, or 50 billion kg. or 50 million tonnes. Since estimates of total global stocks of gold run as high as 2.5 million tonnes, or 2.5 million kg, two weeks global gasoline usage provided by gold (at current prices)would exhaust the world's gold stocks.
So clearly, the price of gold would have to be much higher. How much higher? Well, if it were 20 times higher it would take a year to use up the world's gold, so let's add another factor of 1000. After all, if it were 100 times more, the world's economy would be looking at the end of the gold engine's usefulness in the forseeable future.
Increasing gold prices by a factor of 20,000 pushes current prices to about 2.5 million dollars per ounce, and that is a real problem for using gold as a reserve. The problem here is that gold is now too valuable. After all, with this factor applied to current reserve amounts, the total world monetary gold reserves, between central banks and private reserves a total of about 60,000 tonnes, would amount to 3 tonnes, which is just silly.
Another way to look at this is to equate the world petroleum production with gold production, which is about 1.5 million kg. The ratio works out to about 666,000 gallons/g of gold. At 2 dollars/gallon, that's 1.3 million dollars per gram. This presumes, essentially, that you can run a car for an hour using 1.5 micrograms of gold.
To claim that the increased demand for gold would increase production is almost certainly true, but it's not at all clear what the resulting production would be. For instance, the total gold dissolved in the world's oceans is only on the order of 20 million tonnes, or less than a decade's requirement.
The takeout of all this, I think, is that any widespread gold engine technology which outcompetes petroleum to the point of replacing it would have the effect of long-since depleting the world gold supply to nothing, or eliminating gold as a workable currency basis. Same long-term effect, really.