clippy_dev: Move config parsing to the new parsing framework

[Jarcho]

Based on #16645

changelog: none

[rustbot]

Some changes occurred in clippy_lints/src/doc

cc @notriddle

[rustbot]

r? @dswij

rustbot has assigned @dswij.
They will have a look at your PR within the next two weeks and either review your PR or reassign to another reviewer.

Use r? to explicitly pick a reviewer

Why was this reviewer chosen?

The reviewer was selected based on:

• Owners of files modified in this PR: 7 candidates
• 7 candidates expanded to 7 candidates
• Random selection from dswij, llogiq, samueltardieu

[rustbot]

This PR was rebased onto a different master commit. Here's a range-diff highlighting what actually changed.

Rebasing is a normal part of keeping PRs up to date, so no action is needed—this note is just to help reviewers.