I think that the problem might be that the ‘Posts’ post type (so-called blog posts) does not have a page assigned to it. What happens if you switch to using ‘As an Archive’? Disable Ajax before you test, but if that works (it might not look right, but the correct posts should be there, then that may be the issue.
I do not know how your theme creates templates and pages, but the content on your home page looks static, without any archive loop being run, and thus nothing for our Post Type Archive method to hook in to.
Do you want the results to appear below the search form, or on a new page?
If they are supposed to be on the same page, then using our Shortcode Results Method might be better, as that would at least add an archives loop to the page.
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