Hi!
The ajax response speed mostly depends on the server performance, that is the biggest factor. I have made a few tests, and the average measured speeds were around 0.8 seconds for your server, which is extremely good for a WordPress installation. Check a test recording: https://youtu.be/NsnLyKpHwe0
From the plugins perspective there is not much you can do, the rest is up to the server now. But honestly, results below 1 second are very impressive for an ajax search request.
Sites which have “instant” searches, usually have a 3rd party search server, where the search process is executed on a different server, and the results are sent back to the site. Usually however these solutions have a pricey monthly fee, and are likely a bit harder to implement. Unfortunately that is beyond the scope of this simple plugin.