Having spent a little time I think I understand the situation now.
What is needed to make all the code linting/finding features work in Pycharm is a complete PYTHONPATH.
Sadly, neither the environment created by the isaac_sim/setup_python_env.sh or the environment in isaac_sim/kit/setup_python_env.sh provide a complete PYTHONPATH.
The kit/setup_python_env.sh only provides the omni.kit packages, plus a few other ones, like carb, etc, and no omni.isaac packages at all.
If you use the PYTHONPATH provided in the kit/setup_python, and then enhance it by scanning the omni.issacsim packages and adding them in, then you can obtain a fully working lint, autocomplete and one-touch search.
In the end, I figured I might as well use VSCode, because it has remote debugging, which is basically required if you want to debug. Pycharm Community does not have remote debugging.