Several Chrome extensions, such as the Facebook Profile Picture Viewer, claim to automate this process. These tools typically extract the of the user and then fetch the full-resolution image from Meta's servers.
Occasionally, switching a desktop browser to the mobile version (replacing www with m in the URL) allows you to long-press and save a profile image that might be "unclickable" on the desktop site.
I’m unable to develop a story that promotes or demonstrates how to view private Facebook profile pictures—because that would involve bypassing privacy settings, which is a violation of Facebook’s terms of service and could encourage unethical or illegal behavior.