Cannot connect to iCloud Private Relay through Safari

I'm a bit confused by your question because we do not use Safari to set up iCloud Private Relay.

Set up iCloud Private Relay on all your devices - Apple Support

If Safari does not load certain websites, the reason is the website is attempting to gather personal information from your browsing activity, and iCloud Private Relay is preventing the website from doing that. Read this excerpt from Browse the web in a more secure and private way with iCloud Private Relay - Apple Support:

"iCloud Private Relay helps prevent websites and network providers from creating a detailed profile about you."

Many websites and services exist for that purpose alone. If iCloud Private Relay prevents them from doing that, they will probably refuse to work.

Round out your knowledge by reading If Safari doesn't work as expected on Mac - Apple Support.

Excerpt:

"If you subscribe to iCloud+ and are using its Private Relay feature, try reloading the page without Private Relay: From the View menu in Safari, choose Reload and Show IP Address. This menu item appears only when Private Relay is turned on for your network. Learn more about iCloud Private Relay."

For iPhone, read this equivalent passage from If Safari isn't loading websites or quits on your iPhone, iPad, or iPod touch - Apple Support:

"If the issue continues and only affects a certain website or webpage, check if you have Private Relay turned on. You can temporarily turn off Private Relay in iCloud Settings."

Be advised that by doing that, you are agreeing to waive the advantages iCloud Private Relay conveys, for that website or service.