Hi!
Thank you for the details, it helps us a lot!
Unfortunately the link to the page, where the plugin should be visible is broken. I am not entirely sure, what exactly are you trying to achieve yet. I believe, that the best possible implementation always depends on the situation. Usually, using the plugin shortcode on a page is the best way.
You can also enable search results page override, so when redirecting to the results page, the plugin will try to push it’s own result to the results page template.