Why don't commercial aircraft use all-movable tail surfaces like my RC plane?
What is better and easier for small-scale models is not necessarily better for larger aircraft.

First, you can't say that the fixed part "does nothing". The tail is primarily a stabiliser; without it, a normal airplane will not fly at all. Only then it is a control surface, which allows it to fly well and how you want it.

Consequently, the size of the tail is primarily driven by stability considerations. For control, you need a certain amount of it and not too much more. If you have an all-moving tail, you may find that you need to move it only very slightly, or else control will become too sensitive. This is not necessarily easier to do.

From design point of view, the all-moving tail requires that the whole stabiliser was attached to a single shaft. On a small RC plane you can afford to make it of a simple wire. On a large aircraft, this translates to something bigger than your suburb water supply pipe: stabilisers of planes like A380 or An-124 are larger than wings of some airliners. Apart from the scale problem, this makes it a single point of failure, which makes it still heavier to be reliable.

Another design problem is nice aerodynamic coupling between the moving tail and the fuselage. Often designers have to invent something like this (A330):


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Overall, it is usually easier and lighter to make a conventional tail and attach an elevator (and rudder) to it.

Nevertheless, quite many airplanes have an all-moving stabiliser. There are several different reasons for this.


To accommodate a large range of centre of gravity (CG) locations, i.e. load distribution, and/or different aerodynamic configurations (e.g. flaps). This requires a large amount of control. But this control is typically 'slow': the load or configuration doesn't change in fractions of a second. So the tail is split between the slow all-moving part providing the so-called trim, and the fast conventional elevator. Most airliners have such arrangement, like the A330 from the above picture. But the largest ones, the An-124 and -225, don't use it: too hard. They entirely rely on the elevator.

There is rarely such a demand for the vertical tail: airplanes are more or less symmetrical, so rarely will you find an all-moving vertical stabiliser like this.
To provide easier load balance for small airplanes with reversible control, i.e. those where the control surfaces are moved by the pilot's muscle force. If you make an elevator on normal hinges, it may be quite hard to move it at high speed. Various tricks are used to make the force comfortable (which should be 'just right', not too much, not too small). But one of the straightforward solutions is to make the whole tail move about a carefully chosen axis. For smaller GA aircraft, the scale problem is not too bad yet, so many designs use this approach, for example, Piper PA28


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(The thing you see at the trailing edge is not a traditional elevator, it is the anti-servo tab that provides the 'just right' force).
Normal elevator control efficiency. Sometimes, airplanes just need more control than the normally-sized tail can provide. This primarily applies to agile supersonic aircraft. There are several reasons for it:


Agile manoeuvring obviously needs more control;
At supersonic speeds, pitch stability (and thus control effort) increases significantly;
Elevator efficiency falls. The thing is, at subsonic speeds, the normal elevator redistributes the airflow over the tail such that more air is affected than just the area of the elevator assumes. The relation is $\sqrt \frac{S_{elevator}}{S_{tail}}$, so if you have elevator occupying half the tail, its efficiency will not be just $0.5$, but $\sqrt 0.5 \approx 0.7$ compared to the all-moving stabilator deflected to the same angle. But at supersonic speeds this doesn't apply, and the relation is just linear. So all-moving surfaces make more sense. Occasionally, you can even find an all-moving vertical tail, like on Tu-160 (note it has all-moving horizontal tail as well):



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