Auto merge of #50541 - QuietMisdreavus:rustdoc-errors, r=GuillaumeGomez

Auto merge of #50541 - QuietMisdreavus:rustdoc-errors, r=GuillaumeGomez

rustdoc: replace most (e)println! statements with structured warnings/errors

Turns out, the rustc diagnostic handler doesn't need a whole lot of setup that we weren't already doing. For errors that occur outside a "dealing with source code" context, we can just use the format/color config we were already parsing and make up a `Handler` that we can emit structured warnings/errors from. So i did that. This will make it way easier to test things with `rustdoc-ui` tests, since those require the JSON error output. (In fact, this PR is a yak shave for a different one where i was trying to do just that. `>_>`)