why didn't the Wright brothers go for a gas turbine?
Simple, because the material technology required didn't exist. It was however, up to letting the Wrights build a petrol engine with an adequate power to weight ratio to permit flight.
I have a book printed in 1912 which contains a diagram of a gas turbine but its of a massive steel construction just like those steam turbines in the previous post. It took a great leap of imagination (and the development of Nickel steel alloys) to think that that gas turbine could become a jet engine and that's what Whittle had.
Going back to the Wrights, it always surprised me that two bicycle manufacturers didn't use their bicycle wheels on their aircraft especially when Sir George Cayley had invented them (tensioned wire spoked wheels) for HIS aircraft.