Hi,
Thank you very much for the details!
The main difference with the two implementations is, that the other plugin actually uses it’s own results container (not actually filtering the Elementor widgets), which is very different.
If you do the same with Ajax Search Pro, the results will be populated just as fast or even faster. The reason is, that displaying results via an ajax request to the WordPress ajax handler is much faster, as the front-end does not have to be fetched.
If you want to replicate something similar to the other plugin output, maybe something like this example, then you can use the auto populate feature to initially display results in the original plugin results container – so not connected to an Elementor widget.
On a different note, I think you may have some issues on the site front-end, it takes over 11 seconds to load any front page. I measured, because this reflects on how fast the plugin can update the elementor widget with the new results, it has to wait for the server to respond. The actual search process takes like 15 milliseconds, while the page loads for over 11 seconds. The server seems to be fine, the back-end works perfectly and responds super fast, there is something on the front-end delaying the loading.