I'm reluctant to accept this change as get_sphinx_doc_url is a public function that other modules can use to query documentation URLs. I'd like the caller to still be able to catch the ValueError and display the message to the user themselves.

If you're wanting to make the warning on line 197 more specific -- to say why the documentation URL couldn't be obtained -- how about including the exception message text? E.g.:

except (ValueError, requests.exceptions.ConnectionError, requests.exceptions.Timeout) as e:
	# Couldn't get it from PyPI, trying fallback mapping
	if project_name in fallback_mapping():
		doc_url = fallback_mapping()[project_name]
		intersphinx_mapping[project_name] = (doc_url, None)
	else:
        stderr_writer(f"WARNING: Unable to determine documentation url for project {project_name}: {str(e)}")

I understand your point, removing the raise was here to ensure all URLs are queried until one is good.
Due to the addition of the home-page URL in docs_urls, they may be more than one URL to iterate over. Keeping the raise makes get_sphinx_doc_url always return or fail on the first one, so if the documentation URL is bad, but homepage is good, it will not work.