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Many conventional RC planes use control surfaces at the tail just like the image below:

You can see there is a part in front of each surface control that never moves and only the surface control itself moves.

So, I built an rc plane without that part in front of the surface control. Take a look below:

This plane I built flies really well (no aileron). So I was wondering: why commercial aircrafts or even rc planes dont use this type of rudder/elevator without the part in front of it? See: making the control surface the way I did reduces drag since you dont have a fix part at the tail that does nothing, it reduces weight cause you just need the control surface and nothing more and it is also way easier to build. I cant understand why commercial aircrafts and rc planes dont make use of this exactly same type of rudder/elevator!

Could you please tell me the reason?

Well they do to a degree.

Large commercial aircraft have considerably larger CG ranges than their smaller counterparts do in order to accommodate a wide range of loadings as well as shifts in the Cp due to compresssibility effects at high Mach numbers and changes of AoA. To accommodate this, most airliners make use of a full moving tailplane for pitch trim, thence an additional elevator control surface is added for pitch authority by the pilot. This offers a compromise between excellent pitch and minimal drag and weight as opposed to actuating the entire tailplane. Now fully movable tailplanes for pitch control have been implemented before, as Ron Beyer points out in the L-1011. But it’s not very common.

For a small RC plane that travels at slow speeds your design would work great. However these type of designs have structural limitations. For an all movable vertical and horizontal stabilizer setup all the forces exerted on the control surfaces would be transferred to a movable bearing. This would be a single point of failure if the bearing were to fail. Normally the vertical stabilizer or horizontal stabilizer would use several bolts to attach it to the airframe. In some airplanes these control surfaces are welded on. This distributes the stresses more equally. I fly a Piper Cherokee with an all movable horizontal stabilizer (stabilator) and one of our pre-checks is to make sure there is no give in the bearing. If the bearing was to give out we would have a bad day.

Some do, for example the entire elevator (on the tail) moves on an Airbus A330

The fixed tail surfaces are to provide a passive weathervaning effect for stability. If you have an all-flying tail, it tends to just want to trail into the airflow, so you need one of two things; hydraulic control, where the control surface is rigidly positioned by actuators and the control feel is by springs (The F-86 had an all flying hydraulically operated horizontal tail), or if non hydraulic, the surface needs an anti-servo tab (like the C-177 and just about every cable operated stabilator tail out there) to provide a speed sensitive restorative force that drives the surface to a certain position hand holds it there, and to provide a desirable stick force gradient (increasing force with displacement from trim speed). The anti-servo tab also provides the pitch trim function by changing its neutral point when you adjust the trim, changing the airplane's trim speed.

There are some gliders that have all flying tails with no servo tabs, and these depend on a bungee spring in the elevator control circuit for all of their stick free pitch stability. I used to fly a glider called an LS-1 that was like this, and the bungee wasn't particularly stiff, and the airplane was divergent in pitch if I let the stick go, but it was fine if I was holding it.

And airplane like the Cardinal or Cherokee, with a manually operated stabilator tail, would be pretty difficult to control without the anti-servo tab. The tab's control rod is a very critical inspection item on a pre-flight. The principal advantage of flying tails is more control authority for a given surface area (the Cardinal has MASSIVE tail power compared to similar aircraft), and the ability to fine tune the pitch feel gradient by how the anti-servo tab is geared.

Your RC airplane gets away with no anti-servo tab because the electric servos hold the surface at a given position. Basically the same as hydraulic
controls.

The Zenith family of homebuilts use an all-flying rudder with no fixed vertical fin. This wouldn't normally work so well without a fixed fin to provide a passive weathervaning force, but the Zenith gets away with it by having a very large fuselage vertical surface area aft of the wings. Otherwise it would have to have an anti-servo tab for the rudder in the absence of a vertical fin.