rustdoc: Pass Context as a &mut to allow to remove RefCell fields

[GuillaumeGomez]

It was discussed in #90183 because of the add of the Context::deref_id_map field.

[Milo123459]

@rustbot claim

Going to give this one a try!

[Milo123459]

To be clear, I'm removing the RefCell from the types, not removing those fields, right?

[GuillaumeGomez]

Yes.

[jyn514]

Hi @Milo123459, have you had time to work on this? Feel free to ask here or on Zulip if you need help :)

[Milo123459]

I have worked on it pretty actively. The amount of borrowing errors is.. indescribably large.