Rollup merge of #131945 - aDotInTheVoid:footnote-time, r=notriddle

rustdoc: Clean up footnote handling

Best reviewed commit by commit.

Extracts footnote handling logic into it's own file (first commit) and then makes that file slightly nicer to read/understand.

No functional changes, but lays the groundwork for making more changes to footnotes (eg #131901, #131946)

Author: Zalathar