Reply To: Does the indexing also support the "s" search?

It will execute the same query for both the normal and the results pages. The live search is usually a bit faster, as WordPress does not have to load up all the front-end resorces, scripts, styles, most hooks etc..
The search query is the same, if there is slowness, then it’s more likely due to an overhead. If you want I can debug it in place via FTP access though.