Separate doctests into a separate Nox session and GitHub workflow?

I'm wondering if it would be worthwhile to have a dedicated Nox session and GitHub workflow to check that doctests are working. At the moment I think it would be beneficial to do this, but it'd be worth discussing this further before attempting to implement this.

Advantages:

• We'd be more likely to notice if the doctests are not being run (see #2834)
• We could more reliably ensure that doctests are not counted when checking code coverage (so that we can identify lines of code that the test suite itself does not cover)
• We could shorten the time it takes to run the test suite. I'm not sure if the improvement will be negligible or noticeable, but I'd guess maybe a few seconds?
• We could have somewhat more computationally intensive examples without lengthening the overall time it takes tests to run.
• With pytest-doctestplus enabled, we could have a Nox session that would check and optionally autofix docstrings. When doctests fail, we could print out the command to run to perform autofixes.
• We could more easily isolate problems with doctests.

Disadvantages:

• When running Nox sessions locally, we'd have to run more than one to run test.
• We'd length the list of checks that get run in each PR.

Begin by reviewing the existing Nox sessions and GitHub workflows, then inspect issue #2834 to understand the current doctest coverage gap. Compare the trade-offs in this proposal, including pytest-doctestplus autofixes and coverage isolation; done requires an agreed design before implementation can be scoped.