Why do most propeller-powered planes not have an autothrottle?
Because each jet engine only has one lever; each piston engine has three.
Jet engine's throttle is pretty straight forward - move forward, more power. The computer automatically manages fuel flow to prevent surge, flameout and N1 overshoot.
On a piston engine you have three - a mixture lever, a propeller lever and a throttle lever. Granted, I don't think auto-mixture would be too hard to implement. Now, do I want a high-speed climb or high-power climb? And remember, the prop and throttle must be managed together. You cannot go beyond a certain power range with the props set for low rpm, otherwise you'll damage the engine.
It's like a car's automatic transmission, except it has one more dimension.
Power affects pitch
When you change power you must change the aircraft's pitch altogether. Autopilots on modern airlines have quite a number of vertical modes: ALT, V/S, G/S, VNAV, FLCH, TO/GA, CLB etc. None of those have the concept of gear ratio, which is required for propeller management.
Can you design an autopilot system such that it functions on propeller engines? Absolutely. But it is going to be complicated. Not only the usual operations, but the failure modes, have to be engineered and learnt by the pilots. How do you deal with an auto-mixture failure? RPM mismatch? Such training would add unnecessary cost to an otherwise simple aircraft operation. If your airplane does not have a variable pitch propeller, it is probably too simplistic for this kind of autopilot anyway.
Jets have another property which makes auto throttle a logical choice - the faster you fly, the more powerful the engines. Thus, on a jet, you would have to make continuous throttle adjustment if you change your airspeed. So an auto throttle would be helpful. Pistons do not behave like that.
Turboprops
One of the main reasons for choosing piston aircraft is - it is simple. It just fly. No fancy stuff. No lengthy preflight checks. No computers. I just want to get flying. This thinking also extends to turboprops.
It's worthy to note that the ATR does not have auto throttle as well. The ATR 42 has power management (but not speed management), where activating it will simply move a bug on the torque gauge; the pilots have to manually move the levers to follow it. The ATR 72 has a notch for the pilots to place the throttle levers, but still, no motor to move the levers automatically. I can't think of any reason behind this design philosophy besides simplicity and cost.