But if a patent application includes a method or system that is implemented by a computer, the patent application does not need to teach how to implement the computer.
Correct, because the improvement claimed is not related to the general-purpose computer itself which is old and known.
Given that the computer is the central component to these methods and systems, how can these patent applications get granted when none of them teaches how the computer is implemented in their applications?
The applicant does not bear the burden of teaching everything that is old and known "from scratch". In fact, it is frowned upon to include too much detail about what is old and known in a patent specification.
Let's step away from computers for a moment and consider the invention of controlled flight. Did the Wright brothers have to explain how to make an engine for the plane? Not really because by that time the field of engines was well developed and their invention, which related to controlling the direction of flight, was completely orthogonal to engine design and improvements in engines.
That being said, many patents for computer-implemented inventions do include some high-level detail regarding the computer hardware itself. Usually, it is specified that the computer has at least one processor core and at least one memory, etc, and any peripherals that are needed for the invention to function sensibly may be listed also.
Some inventions do relate to the guts of the computer itself, for example, branch delay slots which some computers use must have been an invention at some point. IIRC, in one such case (I don't know if the applicants were truly the first inventors) it was basically ruled that the applicants lacked enablement because their description of the functioning of the computer with branch delay slots was too high-level.
