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Using Adobe Photoshop Version: 22.3.0 20210302.r.49 660cd2e x64 on Catalina.
To replicate the problem:
Create a path, save path and then duplicate path. After that the pen tool stops working correctly on the duplicated path.
I am working with Photoshop 22.3.0 on a Mac. I followed your steps but had no problem altering the duplicated path.
Have you tried reseting preferences what can be solution for weird Photoshop problems https://helpx.adobe.com/photoshop/using/preferences.html