While reading the release notes of Safari 26.2 which implements support for hidden="until-found", I realized something about this library.
The approach used in version 3 could easily add support for search in the content of tab panels, by using hidden="until-found" instead of hidden and the beforematch event to update the selected tab and hide the current panel.
I have no idea how the approach used in version 4 (using shadow DOM) could be made compatible with this search behavior.