Investigate moving bound dependencies into their functions' files

[JoshuaKGoldberg]

Overview

Idea from looking through #732: processLogger needed to be changed to lazily create the write stream... but isn't even used by the Node API. Although it'd be very hard to not import the file at all for the Node API if everything goes through src/index.ts, it'd still be nice to not set up its dependencies unnecessarily.

Actual Behavior

All dependencies are set up in a single src/api/dependencies.ts file right now.

Expected Behavior

Can each function's file import its own dependencies set up and export the function's dependencies object?

[JoshuaKGoldberg]

Alas, I have again lost interest. 😄