Safari is not connecting to public Wi-Fi on my MacBook Pro

Hmm, sounds like a configuration setting that needs to be checked in Safari.

• From the Safari menu bar, go to Safari > Settings > Advanced > Proxies > Change Settings ...
• On the window that appears, be sure that none of the options are enabled.

Also, is your Mac configured to use a VPN? If so, is the VPN client configured differently when at home vs. when you are at a remote location?

I suggest that you create a new user account on your Mac. The next time you need to use a public Wi-Fi, log into this new account and see if Safari allows you to access the Internet.

If it does, then there is something amiss with your current account that needs to be troubleshot what the cause may be. From what you posted, it still looks like the issue is primarily with Safari itself.

FWIW, I personally use Brave myself. Although, I will occasionally use Safari. I will take a look a closer look at the various Safari settings, but the only one that came to mind was the proxy settings ... which may have been altered when you installed a third-party VPN.

One other thing I can suggest it to run an EtreCheck report, and optionally post it here, if you need assistance with interpreting its results. EtreCheck greatly aides is identifying "difficult" issues with Macs and is recommended many others here.

Ref: Using EtreCheck to Troubleshoot Potential Mac Issues

Great troubleshooting! Ah yes, Private Relay is a VPN-like service that's part of Apple's iCloud offerings ... and it is designed, specifically, to work with the Safari web browser.

Ref:

• About iCloud Private Relay - Apple Support
• Use iCloud Private Relay on Mac - Apple Support
• Apple iCloud+ Private Relay - Explained! - iMore