To answer your question "Why do relational databases index instead of a separate module/process?", you can see this quote from [2] that you linked: "it is clear

To answer your question "Why do relational databases index instead of a separate module/process?", you can see this quote from [2] that you linked: "it is clear when you can use index for sort MySQL will be a lot faster." Relational databases are very fast when they can use indexes.