Add a simple integration test

[AA-Turner]

No description provided.

[hugovk]

Looks like it's failing, but the CI is still passing:

2025-04-12 21:10:24,910 DEBUG en/3.14: Run: 'chgrp -R 118 /home/runner/work/docsbuild-scripts/docsbuild-scripts/logs'
2025-04-12 21:10:24,912 ERROR en/3.14: Badly handled exception, human, please help.
Traceback (most recent call last):
  File "/home/runner/work/docsbuild-scripts/docsbuild-scripts/./build_docs.py", line 572, in run
    self.build()
    ~~~~~~~~~~^^
  File "/home/runner/work/docsbuild-scripts/docsbuild-scripts/./build_docs.py", line 689, in build
    setup_switchers(
    ~~~~~~~~~~~~~~~^
        self.switchers_content, self.checkout / "Doc" / "build" / "html"
        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
    )
    ^
  File "/home/runner/work/docsbuild-scripts/docsbuild-scripts/./build_docs.py", line 476, in setup_switchers
    switchers_path.write_text(script_content, encoding="UTF-8")
    ~~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/opt/hostedtoolcache/Python/3.13.2/x64/lib/python3.13/pathlib/_local.py", line 555, in write_text
    return PathBase.write_text(self, data, encoding, errors, newline)
           ~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/opt/hostedtoolcache/Python/3.13.2/x64/lib/python3.13/pathlib/_abc.py", line 649, in write_text
    raise TypeError('data must be str, not %s' %
                    data.__class__.__name__)
TypeError: data must be str, not bytes
2025-04-12 21:10:25,008 INFO: Skipping sitemap generation (www root does not even exist).
2025-04-12 21:10:25,008 INFO: Skipping 404 page generation (www root does not even exist).
2025-04-12 21:10:25,008 INFO: Skipping copying robots.txt (www root does not even exist).
2025-04-12 21:10:25,008 INFO: Creating major and development version symlinks...
2025-04-12 21:10:25,008 INFO: Full build done (2m 48s).

[hugovk]

And it was okay for this PR: https://github.com/python/docsbuild-scripts/actions/runs/14415254898/job/40430707193?pr=275

Caused by #280?

[AA-Turner]

#284

[AA-Turner]

This one passed, after #280 was merged.