I changed the demo ap to show timestamps. I can confirm that entries are added to existing sessions after what seems (see below) some delay.
This can be made more obvious by deleting all browser data (cookies etc) so that a new session is forced and the new session is at the top.
Also: the refresh button next to the search box seems to refresh the logs?! I never thought it did very much. So no green box (with “there are newer log entries”) now, but this button refreshes the logs. Closing and reopening the app log screen will not refresh the log entries, but this button will.
So my conclusion for now: user error due to non-intuitive log screen UI.
Learned something today.
Updated demo app: Anvil | Login