I have implimented Super Speedy Search by WPIntense on the site in the private comments.
It is set to product search only and to override the search results page.
The search results generated by AJAX are correct but it appears a setting in Enfold loop-search.php or search.php is preventing the plugin showing the correct results.
I had the same issue with Relevanssi.
To add – It doesnt matter whether you use the Super Speedy Search widgets or the built in Enfold search or the woocommerce search widget.
All of them have the same behaviour.
Something in Enfold is overriding the search results page generation process – Stopping &post_type=product from picking up the correct search results.
Update on this
It appears to be a bug withing Super Speedy Search.
I have raised a bug ticket with SSS so please hold on.
Thanks for the update, we will leave this thread open to hear back from you, I see that you also found that the issue seems to be related to if the user is logged in or out, so that the Super Speedy Search plugin results shown to admins are correct and the results shown to normal visitors is incorrect.
We are limited from working on third party plugin, hopefully Super Speedy Search will have a solution for this, but we will leave this thread open for now.