Does an Airworthiness Directive from the FAA ever correct a process rather than a mechanical component of an aircraft?
First off : yes, Airworthiness Authorities do supervise processes, and may require changes to them.
The ultimate purpose of the aircraft is to be operated, and a "process" is just another word for "operation" : a sequence of actions involving a complete system, or parts of the system.
A well-designed, certified and as-safe-as-can-be airplane can be misused and crashed if the operational instructions are crap. Therefore all operational instructions for pilots or maintenance are part of the certification package of the "aircraft system", and known as Instructions for Continued Airworthiness. And as an integral part of airworthiness, they are subject to scrutiny of the authorities, and any resulting changes.
Also, aircraft design and production processes are certified and monitored by authorities, just like flight ops or maintenance processes.
That being said, there are relatively few issues with processes, and they are usually less dramatic and immediate than issues with the aircraft. Thus most ADs would address issues with the machine itself.
Could a process be regarded as a system?
No. A system is a collection of interdependent things/entities working together to achieve some goal. So as mentioned it is somewhat tangible
As mentioned before, a process is a sequence of actions involving the system.
Think of the difference between "a table" (system) and "eating dinner" (process which may involve the table)
Regarding your initial question, you have to separate the actual root cause of a problem, and how you deal with it and/or solve it.
A problem with the system does not necessarily require a modification to the actual system itself ; it could be solved by adapting the process. And this is usually the case for quick short-term solutions, until the system itself can be corrected (which is generally complex and complicated)
For example, in case of a broken leg on the table, the "eating dinner" process could be amended to avoid using the broken side, or by adding a preliminary step to the process to stabilize the table with a shim. It's far easier to implement than repairing the table, but of course it is much more inconvenient, so a repair would be carried out at some point.
In aviation, dispatch under MMEL may use this idea. Sometimes it is more efficient to fly an aircraft with inop components, even though the pilots may have to slightly modify their flight operation (process) due to extra limitations, for example.
I am not so familiar with ADs, but it seems to be the same. Even though an issue may reside in the aircraft itself, the short-term workaround would be an adaptation to the operations involving the aircraft.
For example, if we look at the P&W GTF issues, the various problems with the propulsion system were initially addressed by operational instructions (extension of engine warm-up/cool-down times, ETOPS limitations, total grounding in India, etc...).