Rover also looked at GTs for it's cars in the same timeframe, they even raced some at Le Mans which did reasonably well.
In both 1963 and 1965, they were the fastest British car in the race. I'd call that a little more than reasonably well, honestly. I've wondered why Rover, BRM and Rolls-Royce after 1965 didn't drop the gloves and build something to hunt the Ford GT40 and Ferrari 330, particularly with Jaguar's XJ13 on the drawing board by then. The addition of regenerators to the Rover-BRM made the thing have shockingly-good fuel economy for a car, and if one was to add diesel into the equation, you could have had a shot at winning it all.
Colin Chapman also raced GTs at Indy, or tried to I think.
No, they did race it. The Lotus 56, which used a Pratt and Whitney gas turbine engine, was a monster - 500+ horsepower was good, but the 1100+ foot-pounds of torque was even better. The thing had so much beans that the 56 needed its four-wheel-drive even at Indianapolis. Joe Leonard was running away with the race when the engine's torque snapped the fuel pump drive shaft with nine laps remaining. GTs are fantastic for Indianapolis, because the motor is at high revs all the time, which reduced the problems with fuel consumption.
As others have said they work well at constant speed, prefereably revving their tits off, and when they are at idle they use just as much fuel as when they are at full revs which sucks when your sitting at the traffic lights. This is so bad that GTs were impractical as a passenger train engine in the 60s due to the fuel use in stations.
If we're looking at the locomotives here, I have to bring up these:
The top one is the three-unit Union Pacific "Big Blow" gas turbine, to this day still the most powerful single locomotive to even run on American rails at no less than 8500 horsepower. Union Pacific calculated that on a flat track one of those turbines could move more than seven hundred freight cars, and they regularly moved trains of 13,000+ tons on UP's transcontinental main line between Odgen, Utah and Council Bluffs, Iowa. The only reason these engines were retired was fuel costs. They had been designed to use Bunker C fuel, which is extremely heavy fuel oil that the tenders had to heat up to 200 degrees Fahrenheit before the locomotives could use it, and new uses in plastics and other industries made the fuel more expensive and doomed the engines.
The second engine is even crazier. It was Union Pacific and General Electric's attempt to fix the problem of fuel costs by running it on coal. It looks like a Frankenstein because it is, effectively, an Alco PA passenger diesel front, a Great Northern W-1 electric locomotive in the middle that had had its guts torn out and replaced with a gas turbine and a Big Boy tender with the water tank replaced with a coal crusher. It had maintenance problems right from the start, but when it worked, it worked very well indeed - 7000 horsepower delivered over 12 powered axles, which would undoubtedly mean immense tractive effort.
All UP gas turbines combined diesel start engines with the turbines, with the goal of the big engines getting the train going before the turbine cranked up and got it going considerably faster. The maintenance problems were primarily related to soot building and corrosive exhaust causing the turbine blades to erode, problems which the coal burner magnified. I've wondered honestly if anybody ever thought of trying to use these things on natural gas instead of heavy fuel oil or pulverized coal.