Add an equivalent of "--check" on rustdoc

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.