With the release of Rust 1.74, there are some new or modified clippy
lints that need adaption in the code.

The main change was the removal of the `private_in_public`.
https://rust-lang.github.io/rfcs/2145-type-privacy.html

Then two more changes were required, in one case to adhere a lint and
the other to allow it. When talking about what an "application" needs to
do when setting up `tracing-error`, it makes sense to include `fn
main()` in the doctest, even though the lint recommends against it.