Hi Tom,
It checked the page, and both of the widgets were still present, so I only removed the plugin class name from the posts widget, so it does not override, and only takes the archive widget into consideration. You should probably remove the old posts widget, I guess you don’t want that there anymore, I did not want to remove it, just in case you need it.
I also changed the pagination feature to numberic, and that seems to have solved the issue.
One thing I noticed, that when I am not logged in, there are many many errors on the console log, which affect the overall page load events and terminates the script execution. That is very likely the cause of the malfunctioning search bar. Make sure investigate all the errors and fix them one by one, that will resolve the issue with any non-responding feature.