I spend a fair amount of time thinking about airplanes. I don't fly much, mostly because I'm too cheap. When I fly commercial I like to complain about how uncivilized the whole ordeal is. When I fly in private planes, I complain (privately, to myself) that these things are noisy, expensive, inefficient antiques. In spite of that, I really like airplanes, all kinds of airplanes, and if I could ever get the kind of airplane I want (six passenger Cadillac, 400 MPH cruise, 5 GPH, buck and a half price tag) I'd buy one.
Airplane builders like turbine engines. I can understand why, they are pretty cool. They have a great power to weight ratio, which is the aircraft designers holy grail. They have only one moving part. They are simplicity itself, unless you try to build one, then you run into all kinds of problems, like complex shapes, high temperatures, difficult to work exotic alloys. They aren't something you can build from scratch in your garage. And then there is their high fuel consumption. That can be a bit of a problem, to put it mildly. So unless you are made of money, turbines are right out of the running for a private plane, which means you are going to be using a conventional piston engine.
When you are using a propeller to drive your aircraft, bigger is better, as in more efficient. Making them too big results in other problems, like ground clearance when taxiing. It also means they need to turn slower. You don't want the tip of the propeller to exceed the speed of sound or bad things happen, like they quit working, or your ears turn to mush. So if you are using a high speed engine to drive your prop, you need a speed reducing gearbox.
Gear reduction is a bit of a bug-a-boo. All the little Cessna and Piper single engine private planes you see at your local airport typically use large displacement Lycoming or Continental engines that drive the propeller directly, that is, without any gear reduction. It makes the powerplant simpler mechanically, but it's become a bit of a dead end. All these engines were designed a thousand years ago. Solid, reliable, but very much a niche market, with corresponding niche market prices.
Because of the prices, some experimental airplane builders have turned to using automobile engines. Automobiles have always used gear reduction, so their speed (RPM - Revolutions Per Minute) has not been constrained like the airplanes, but it means that, like the turboprop, you need some kind of gear reduction. This adds weight and complexity, but it also means you get a powerplant that costs less than half what a conventional Lycoming or Continental powerplant costs, and that can mean a chunk of change.
Funny thing is, military propeller driven planes have almost always used gear reducers. All those big radial engines you see on WWII bombers, and the big V-12's that were used in fighters, all had speed reducers built into them. The engines of that time were still big, slow turning monsters, but with those big ten to fifteen foot props, you still had to cut your RPM's down to something the props could handle.