You need to go into the Extension preference page for the extension, and hit the "Allow Access" permission box for allowing access to web url/link information. I can't remember exact wording, but should essentially be one option here and you have to enable it so that it has permission to pass along URL.
This should be the expected behavior when pressing the button on a new Safari tab. Otherwise, should redirect to the URL your current open tab is displaying.
Then something in the UX is broken. From your store images, I understood that clicking on that tab menu item, it would display last archive date and ask to rearchive, no?
There is a reason why most Safari extensions apps do not just show an icon and a label, but actually explain how to use their embedded extensions.
The extension is open source, I'm open to new ideas on how to improve this further. The current Edge/Firefox extensions just error out, and this was not acceptable for this version of the extension per the AppStore rules.