rustdoc: Automatically index crates (#13014)

This PR removes the need to use `/rustdoc --index <CRATE_NAME>` and
instead indexes the crates once they are referenced.

As soon as the first `:` is added after the crate name, the indexing
will kick off in the background and update the index as it goes.

Release Notes:

- N/A

crates/rustdoc/Cargo.toml

@@ -23,6 +23,7 @@ heed.workspace = true
 html_to_markdown.workspace = true
 http.workspace = true
 indexmap.workspace = true
+parking_lot.workspace = true
 serde.workspace = true
 strum.workspace = true
 util.workspace = true