Vlad Hi Vlad, hope this video helps, I hope the privacy blur doesn't obstruct too much. It does look like the "auth success redirect" is failing the first time, but not the second.
It's not shown in the video, but refreshing the page on the sign-in prompt does not make it go away. It's only if I log in again or search from the url bar again that I can start searching.
I don't currently use iCloud Private Relay.