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(Inspired by this question about rudder hand control on joystick-equipped aircraft.)

Most civilian fixed-wing aircraft (post-1987 Airbus airliners being the primary exceptions) use a yoke (essentially a steering wheel mounted on a vertical column that can swing back and forward) to control pitch (pushing the yoke forward pitches the nose down; pulling the yoke back pitches the nose up) and roll (rotating the yoke clockwise rolls the aircraft to the right; rotating the yoke counterclockwise rolls the aircraft to the left), but control yaw via a separate set of rudder pedals (pushing on the left-foot pedal yaws the nose to the left; pushing on the right-foot pedal yaws the nose to the right).

If the yoke were used to control yaw as well as pitch and roll, this would allow the pilot to make coordinated turns using just their hands, rather than having to remember to push with one of their feet at the same time, and would eliminate the risk of accidentally applying the brakes when steering on the ground.

A couple of possible ways for yoke-based rudder control suggest themselves; one would be to tilt the column from side to side (tilting the column left would yaw the nose to the left; tilting the column right would yaw the nose to the right), while another would be to push one of the yoke's horns forward while pulling the other back, rotating the yoke about its vertical axis (pushing the left horn forward and pulling the right horn back would yaw the nose to the right; pulling the left horn back and pushing the right horn forward would yaw the nose to the left).

Here's an illustration of what I've in mind:

Why don't any aircraft use the yoke to control all three axes, rather than just pitch and roll?

The modern control yoke is directly derived from the "joystick" control that became standard on aircraft in the days when Glenn Curtiss personally ran the company that was the main competitor to the Wright brothers.

After inventing aileron control (the Wrights were still using wing warping at the time -- this was before 1910), Curtiss needed a way to control movement of the ailerons, and subsequently of the rudder. The original 1903 Wright Flyer had the wing warp controlled by sliding the pilot's platform (a flat surface, on which the pilot lay prone) right and left, and coupled the rudder, so that roll and yaw were inseparable. Curtiss decoupled them, and needed to add a third control -- and since he was also sitting upright, even in his first airplane, his feet were available.

Running the elevators and ailerons on the control stick was obvious, and it was equally simple to put one's feet on a bar that directly operated the rudder -- and this layout became the standard almost instantly. Even the Wrights adopted it before they demonstrated their Flier to the Army.

Over time, there have been a few examples of variations. Airplanes that brought back coupled rudder and aileron, like the Ercoupe, let the pilot fly with "feet flat on the floor" -- and it seems to me there was at least one design, from the biplane era, of a transport aircraft with rudder operated much the way you describe; a control wheel mounted on a joystick, with stick movement controlling roll, and wheel rotation controlling yaw.

The fact this has only appeared in a very small number of designs suggests that, as noted in a comment, it's a bad idea to change something that's been long standardized -- yet, we have a good number of aircraft, ranging from sailplane to jet fighter and large transport, that use "sidestick" -- in which, in the transport case, the pilot in command actually flies with his left hand, while the copilot flies with his right. Joysticks continue in wide use as well, especially in smaller or higher performance aircraft, or those with ejection systems.

The other, and I believe the main reason we don't see control schemes like what you describe is that it becomes impossible to maintain precise, separate control of roll and yaw. When the same pair of hands are doing both jobs, the brain will mix them together, or in trying not to, will reverse mix (leading to a forward slip or a skid, the latter widely considered very hazardous at low altitude and speed). If you have an aircraft in which it's difficult to avoid mixing either adverse or proverse rudder while applying aileron, it'll be difficult to land or take off in crosswinds, hard to maintain a precise final approach, and nearly impossible to fly high precision maneuvers (like air to air refueling or tight formations).

It's just a bad idea Sean. Believe me if you've done any flying you would NOT like a control column that you have to shove sideways, or a twisty yoke, for rudder as well as roll and pitch. Your feet are sitting there doing nothing anyway. And you need to be able to control it one handed so you can work the thrust levers or power levers or throttles with the other. How would you work such a column with one hand?

Plus, in any transport airplane with a yaw damper system you never touch the rudder pedals once airborne unless an engine quits. And if that happens you'll be glad you have your upper thigh muscles to the do the work of holding in rudder input for an extended period, and not your forearms already busy with 2 other jobs.

I'm imagining trying to hold a yoke pushed to the side following an engine failure on rotation, while also controlling pitch and roll with it, while my feet sit on the floor being useless... very unpleasant.

Where it might be viable is with a side stick FBW controller where the stick rotates for yaw, like a computer joystick. But even there, I'd rather have my feet do it.

Such designs do not work well when the rudder is required other than during turns.

A typical situation would be the use of rudder to counter the p-factor (asymmetric turning tendency) on propeller aircraft. After take-off, one might need to apply a significant amount of rudder during climb at high-power and high-AOA.

It would be very inconvenient to have the yoke "tilted" or "twisted" during the climb, even when the aircraft is not turning at all.