You’re assuming private relay is like a VPN. It’s not [1]. Apple can see your IP address but not the service you are connecting to. The independent private relay partner (eg cloudflare) can see the service being connected to but not your IP address. The website can see nothing about the connection (unless you login). This also prevents tracking you across sites btw.

[1] https://blog.cloudflare.com/icloud-private-relay/