On a dedicated sever I don’t think this should be an issue, our tests were conducted on a single core virtual machine with 1G ram only for 100 000 posts.
There might be something else at play here. When I try to visit the default results page without triggering ajax search pro override (simply adding “?s=” to the main URL), I am getting a critical error report on the bottom of the page: https://i.imgur.com/MFugMhI.png
This may indicate something, maybe a memory limit issue in general, or simply a programming error or a bottleneck somewhere. It does not trigger every time, so that may indicate only a memory limit problem. First, try to increase the PHP memory limit a little bit to see if that error disappears permanently.
After that, if there is no change, the you can add temporary FTP and back-end access, and I will try to debug throught the plugin code to see if I can find anything.
First, I will programmatically disable the actual search query within ajax search pro, to see if that makes any difference when the results page loads. If not, then the issue is caused by something else, but if it is faster, then I will go through the code line-by-line to see where it’s stuck.