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:
    [email protected] 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.