Add an equivalent of "--check" on rustdoc

[GuillaumeGomez]

In a recent conversation, it came up that cargo check wasn't running rustdoc lints. I then realized that there was no easy way to just run an equivalent of cargo check with rustdoc to get those lints. So the question is mostly: how should we do it? Extend cargo? Add an option on rustdoc and then call it through cargo?

cc @rust-lang/rustdoc

Also tagging @yaahc who brought up the issue.

[jyn514]

Another thing mentioned is that it's not clear which lints belong to rustdoc and which belong to rustc. If we added doc:: in front it would be more clear the lints are only run with rustdoc.

[jyn514]

Added in #78984