ci: Switch cargo-public-api-crates to cargo-check-external-types

[tottoto]

Motivation

cargo-public-api-crates seems to have version conflict at the dependencies.

error: failed to compile `cargo-public-api-crates v0.2.0 (https://github.com/davidpdrsn/cargo-public-api-crates#13bf827a)`, intermediate artifacts can be found at `/tmp/cargo-installDFx8K1`.
To reuse those artifacts with a future compilation, set the environment variable `CARGO_TARGET_DIR` to that path.

Caused by:
  rustc 1.80.0-nightly is not supported by the following package:
    indexmap@2.12.0 requires rustc 1.82
  Either upgrade rustc or select compatible dependency versions with
  `cargo update <name>@<current-ver> --precise <compatible-ver>`
  where `<compatible-ver>` is the latest version supporting rustc 1.80.0-nightly

https://github.com/tower-rs/tower-http/actions/runs/18697143413/job/53317419830

Solution

Switches cargo-public-api-crates to cargo-check-external-types.

[jplatte]

As of #619 we're using my fork of cargo-public-api-crates. Do you think it still makes sense to switch?

[tottoto]

I think a benefit of cargo-check-external-types is that it handles internal crates like pin-project-lite properly.

[jplatte]

That's actually a somewhat recent bug that pin-project-lite is found as a public dependency by cargo-public-api-crates.. I think.

Anyways, we can switch. I also don't need to maintain cargo-public-api-crates anymore then :)

[jplatte]

Mind doing the same for axum?

[tottoto]

Sure. Opened tokio-rs/axum#3576.

[tottoto]

I noticed that I posted #613 (comment) with a copy and paste mistake: the place of cargo-public-api-crates in the post was intended to be cargo-check-external-types. I fixed it.

[jplatte]

Yeah I figured you meant it that way ;)