Hi Simon,
Well, for that the results page template needs to be customized – or perhaps you can consider using an elementor posts or products widget, as the plugin now supports integration with them.
The only thing I see wrong with that URL, is that the redirection is probably not set incorrectly, make sure to change the magnifier and return button events to redirect to the search results page.
The indexing is automatic, if the index table is used – upon creating new products they are automatically added to the index. You can additionally enable the index table cron job, that will periodically check for new items.