Why are landing gear retraction mechanisms mostly based on the four bar system?
(wikimedia.org)

The schematic above (with four arcs that are easy to trace) will hopefully serve as a starting point. This is a rather simple mechanism, say compared to the A380's body landing gear.

The principle can be explained by trying the proposed solution, and see what problems we run into. If you place only one actuator to rotate the landing gear at the attachment point* to the airplane (magenta colored circle), you run into many problems:


No locking mechanism
Small lever arm
Bigger actuating force needed
Limited movement (no full retraction/extension possible), unless


Packaging the mechanism in a small volume is not required
More linkages are added.



End result: back to square one, the first 3 problems, and no space for it in the limited volume of wings/bellies.

* If you place the actuator in a more convenient place like near the wing root, then it won't be able to get out of the way to stow the landing gear, unless you have a very thick unrealistic wing.

Note from the above schematic that the hydraulic cylinder (1) needs to retract/expand only by a relatively short distance to accomplish the retraction/extension.

The flaps and slats are also complex, not only you need to turn something around an axis, you need to translate it as well, creating the gaps seen, and package it neatly in a small volume.

If this basic objective is clear now, the next step is to read more, and to build models and/or play with kinematic simulation software. Try to build the same mechanism with one link, and then try to solve the issues you run into.