WARNING: pip is being invoked by an old script wrapper. This will fail in a future version of pip. Please see https://github.com/pypa/pip/issues/5599 for advi
I also faced the same problem when I switched to zsh shell from bash.
The solution was simple but I didn't notice it at first.
After I installed pip, I saw this warning
pip is being invoked by an old script wrapper
First I tried this solution
python3.8 -m pip install --upgrade --force-reinstall pip
But then I ran into this issue:

Then I searched how to add PYTHONPATH. I opened my .zshrc and say these lines were commented
# If you come from bash you might have to change your $PATH.
I uncommented the line that followed, and my misery vanished.

Now when I ran,
python3.8 -m pip install --upgrade --force-reinstall pip
Then the warnings of not in path disappeared in thin air, leaving me with a clean output.

I hope this would help anybody who ran into the same problem.