### Body

Same as https://github.com/apache/airflow/issues/33876 but for `publish-docs` command.

### Committer

- [X] I acknowledge that I am a maintainer/committer of the Apache Airflow project.
with run_with_pool(
                parallelism=parallelism,
                all_params=all_params,
                debug_resources=debug_resources,
                progress_matcher=GenericRegexpProgressMatcher(
                    regexp=SDIST_INSTALL_PROGRESS_REGEXP, lines_to_search=10
                ),
            ) as (pool, outputs):
                results = [
                    pool.apply_async(
                        _run_command_for_providers,
                        kwds={
                            "shell_params": shell_params,
                            "cmd_to_run": cmd_to_run,
                            "list_of_providers": list_of_providers,
                            "output": outputs[index],
                        },
                    )
                    for index, list_of_providers in enumerate(provider_chunks)
                ]
        check_async_run_results(
            results=results,
            success="All packages installed successfully",
            outputs=outputs,
            include_success_outputs=include_success_outputs,
            skip_cleanup=skip_cleanup,
        )
    else:
        result_command = run_docker_command_with_debug(
            params=shell_params,