Thank you!
I’m afraid this is not related to the plugin. I completely turned off the queries to see if it makes any difference, but all of the requests are super slow.
Even when I turned off the plugin completely, and run a fetch request to the WordPress ajax handler, it takes over 4 (!!!) seconds to respond to a completely empty request, which does nothing.
See the empty request, and the response times measured. This should be normally sub/around 1 seconds, that is perfectly fine. On the second screenshot you see, that some requests were almost normal, but most of them were simply extremely slow to respond, that indicates a serious problem on the server side. Also note, the search plugin was disabled programmatically during these tests.
It is very much possible, that this is caused by a piece of code or a plugin. I usually recommend deactivating the plugins one by one to see if at any point the response time gets better – usually that one is at fault. Same with the theme.
It can be much more complicated, it is also possible that the server simply can not keep up with the demand, or there is a faulty configuration somewhere.
Firewalls/virus scanners on the server can be also problematic, they will catch all requests, and scan them via heuristics, that can eat up a lot of CPU & Ram.