Fails to run on self hosted Windows runner if pip is not in PATH

Description:

Downstream issue: https://github.com/msys2/msys2-autobuild/commit/fe4bcd08a9fbdd4a2874a67b4491991a76d784a7#commitcomment-153600091

Running the following on a self-hosted Windows runner leads to:

    - uses: actions/setup-python@v5
      id: python
      with:
        python-version: '3.13'
        cache: 'pip'
        cache-dependency-path: 'requirements.txt'
        architecture: 'x64'
        update-environment: false

error:

[error]Command failed: pip cache dir
'pip' is not recognized as an internal or external command,
operable program or batch file.

The problematic thing being "update-environment" (which we need to set, so it doesn't set cmake and pkg-config variables interfering with our builds. I don't mind the PATH, but setup-python only does all or nothing)

It looks like the cache code depends on pip being globally available, while it might not be: https://github.com/actions/setup-python/blob/9e62be81b28222addecf85e47571213eb7680449/src/cache-distributions/pip-cache.ts#L35-L41. This likely just works on the hosted runners since there pip is installed already.

In other places this is handled correctly by explicitly going through the just installed python/pip, for example: https://github.com/actions/setup-python/blob/9e62be81b28222addecf85e47571213eb7680449/src/install-pypy.ts#L177-L182

Action version:
actions/setup-python@v5

Platform:

• Ubuntu
• macOS
• Windows

Runner type:

• Hosted
• Self-hosted

Tools version:

3.13, but any really

Repro steps:
See above

Expected behavior:
The action to run

Actual behavior:
It errors out

Start with the pip cache logic in src/cache-distributions/pip-cache.ts at the referenced lines, then compare how src/install-pypy.ts invokes the installed Python tooling. Reproduce the Windows self-hosted workflow with update-environment disabled and verify that pip caching succeeds when pip is absent from PATH.