CI should fail if public APIs are missing documentation or if the documentation has detectable problems

## What is the idea?

We should build the docs as part of CI to ensure the process succeeds.

`rustc` has a [missing_docs](https://doc.rust-lang.org/rustc/lints/listing/allowed-by-default.html#missing-docs) lint that we could set to `deny`. `rustdocs` also has a set of [lints](https://doc.rust-lang.org/rustdoc/lints.html) that can be set up to emit errors if they're violated. These two changes would help catch mistakes in the public documentation.

These lints can be enabled at as part of each crate in `src/lib.rs` or as command-line arguments to various `cargo` commands, perhaps through the `Makefile`.

## What problem does it solve?

It would be helpful to know when someone has forgotten to add documentation to a public member in a PR or if there are issues detectable by `rustdoc` with the docs that should be corrected.