After fiddeling with different and various installments and versions of Python on MacOS, I stumbled over the article The right and wrong way to set Python 3 as default on a Mac - which works like a charm!
- brew install pyenv
- pyenv install 3.8.7 - the Python version you want to install
- pyenv global 3.8.7 - set it as preferred version
- make pyenv known to your login shell ($HOME/.zshrc)
if command -v pyenv 1>/dev/null 2>&1; then eval "$(pyenv init -)" fi
- use python and pip