Auto merge of #125798 - camelid:refactor-doctest, r=GuillaumeGomez

rustdoc: Refactor doctest collection and running code

This code previously had a quite confusing structure, mixing the collection,
processing, and running of doctests with multiple layers of indirection. There
are also many cases where tons of parameters are passed to functions with little
typing information (e.g., booleans or strings are often used).

As a result, the source of bugs is obfuscated (e.g. #81070) and large changes
(e.g.  #123974) become unnecessarily complicated. This PR is a first step to try
to simplify the code and make it easier to follow and less bug-prone.

r? `@GuillaumeGomez`

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