Fix and modernize GitHub CI Workflow

[nullstalgia]

You can see the results of a successful run here: https://github.com/nullstalgia/eyre-indenter/actions/runs/16245429504

The following lints have been removed from the warn() directive:

private_in_public: Replaced by new, on-by-default, type privacy warnings.

const_err: Deprecated, now returns a hard error.

The missing_doc_code_examples lint being renamed to rustdoc::missing_doc_code_examples means I can't put a global RUSTFLAGS: -Dwarnings, unless I add the rustversion build-dep, but I wanted to avoid adding any dependencies.

At the very least, the stable clippy run does run with -D warnings.

[ten3roberts]

Thanks a lot, this is a great start 💪