Remove `cfg(doc)` and `cfg(doctest)` from `doc_auto_cfg`, fix `not()` making badge disappear

[SpriteOvO]

Context: #43781 (comment)

CC @GuillaumeGomez

[GuillaumeGomez]

Thanks for opening this issue! Someone from the rustdoc team will try to get back to it whenever possible.

[SpriteOvO]

Thanks for opening this issue! Someone from the rustdoc team will try to get back to it whenever possible.

@GuillaumeGomez It doesn't look too difficult. I'm going to try to make a PR for it myself, and I'll update here with or without progress.

[GuillaumeGomez]

Even better! 😄 I'll assign you to the issue in the meantime then.