I'm not much of a GHA expert, so I'd leave changes to that to a separate PR.
(If we do that, we should take the names from the newly-built Intersphinx inventory, rather than parse the .rst. I can do the parsing; the GHA integration is intimidating.)

One extra thing I would like to do is run the script on docs PRs, since a name being documented while also being in ignored_c_api.txt will cause it to fail. So, if someone merges new documentation without updating ignored_c_api.txt, we'll see blocking failures on all C code PRs until we fix it. I haven't looked too closely at the GHA for change detection, but if you see an easy way to do that, would you mind doing it in this PR?

See #143573.

Thanks @encukou for the PR 🌮🎉.. I'm working now to backport this PR to: 3.13, 3.14.
🐍🍒⛏🤖

Sorry, @encukou, I could not cleanly backport this to 3.14 due to a conflict.
Please backport using cherry_picker on command line.

cherry_picker 234a15dc4ec2d8f8ababea91532ebe896a96387a 3.14

Sorry, @encukou, I could not cleanly backport this to 3.13 due to a conflict.
Please backport using cherry_picker on command line.

cherry_picker 234a15dc4ec2d8f8ababea91532ebe896a96387a 3.13

GH-143593 is a backport of this pull request to the 3.14 branch.

GH-143594 is a backport of this pull request to the 3.13 branch.

⚠️⚠️⚠️ Buildbot failure ⚠️⚠️⚠️

Hi! The buildbot x86-64 MacOS Intel NoGIL 3.13 (tier-2) has failed when building commit b06ef5b.

What do you need to do:

1. Don't panic.
2. Check the buildbot page in the devguide if you don't know what the buildbots are or how they work.
3. Go to the page of the buildbot that failed (https://buildbot.python.org/#/builders/1449/builds/1695) and take a look at the build logs.
4. Check if the failure is related to this commit (b06ef5b) or if it is a false positive.
5. If the failure is related to this commit, please, reflect that on the issue and make a new Pull Request with a fix.

You can take a look at the buildbot page here:

https://buildbot.python.org/#/builders/1449/builds/1695

Failed tests:

• test.test_multiprocessing_forkserver.test_processes

Failed subtests:

• test_repr_rlock - test.test_multiprocessing_forkserver.test_processes.WithProcessesTestLock.test_repr_rlock

Summary of the results of the build (if available):

==

Traceback (most recent call last):
  File "/Users/buildbot/buildarea/3.13.itamaro-macos-intel-aws.nogil/build/Lib/test/_test_multiprocessing.py", line 1493, in test_repr_rlock
    self.assertEqual('<RLock(SomeOtherThread, nonzero)>', repr(lock))
    ~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError: '<RLock(SomeOtherThread, nonzero)>' != '<RLock(None, 0)>'
- <RLock(SomeOtherThread, nonzero)>
+ <RLock(None, 0)>