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Summary
Upgrade Sphinx to 5.3.0. This version is available in the Debian archive, but, unlike later versions, upgrading to it would not introduce significant breaking changes. This avoids the pitfalls of #9556. The only change I had to make was disabling table of contents entries for domain objects.
Sphinx 5.3.0 lacks the breaking changes to the gettext builder of the later versions, meaning builder.py doesn't need to be changed at all; it is still compatible with Python 3.8; it doesn't break the favicon; and it doesn't require bumping any of the other dependencies (sphinxcontrib-serializinghtml, etc.).
While #9938 mentions Sphinx 7.4.7 specifically, the issue at hand was that the version of Sphinx used here (4.4.0) was not available on the Debian archive; by switching to one that is, I think we can mark that issue as closed.
Checklist