Fix the PyPI job and ensure the package to-be-sent to PyPI is also tested appropriately

As mentioned in #3976, the PyPI job has been failing for quite some time now, and it is not possible to see if there are issues with only the **PyPI** package that otherwise do not surface in a locally created version. This should be rectified, but as a separate task, not to prevent the merging (and the subsequence working release) of #3976.