When I email myself a link from Kagi search, or copy and paste it to Kagi and try to open the link, I get taken to the log in page and lose the original search. I think when being redirected to log in page, and even in case of log in failure, the source URL needs to be retained.
Perhaps the link could be assigned a unique code opened max 5 times on other browsers, so you don't penalized for sharing a search from Kagi to yourself or someone else.