Thank you very much for the proper details, it helps me a lot!
The issue was, that the default theme search bar was not visible on those pages (home page), so even as the search override was activated, it had nothing to replace there.
Instead of that, I have disabled the override feature (here), and simply placed the ajax search pro search widget to the footer area – which is visible on all pages on your website: https://i.imgur.com/z7PLu2A.png
Now, it should be visible on all pages.
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