Unpin various dependencies

Description

This MR unpins a selection of dependencies, most notably PyJWT which was pinned to an old version due to API changes. As such, this MR also contains a few code changes to address API changes and bugs found by newer type hints in the newly unpinned dependencies.

It might make sense to roll !872 (merged) into this MR.

Currently this doesn't unpin pytest-pylint since doing so caused the tests to hang during test collection for some reason that needs further investigation, and it also doesn't unpin the docs-related pins (Sphinx, sphinx-rtd-theme, and jinja2). It also doesn't unpin SQLAlchemy, since that needs a more significant code change.