Simple default replication worker refactor (#19002)

Logic in this class is going to have to change as part of two big upcoming projects:
- column selection
- progress bars

To prepare for this, I've gone ahead and refactored the run method for readability. This is a monster function. The current function is too long and contains several operational abstractions, increasing unnecessary complexity. This is the core of what we do, so it's important to ensure this code is extremely understandable.

Ultimately we want to probably want to break the run method up into two or more separate classes - one that deals with replication and one that deals with outputs - for better testing, readability and isolation. This sets the stage for that.

I have intentionally NOT removed or touched any logic, nor have I put thought into consolidating the function signatures to preserve as much of the pre-existing logic and keep the changeset small and reviewable.

This changeset only renames and moves code around.