Nice to hear about the site wide search to be implemented in Pulse 5!
Regarding external tools, I have been using FreeFind (paid plan) on three WordPress sites for which WP own site search was not enough (I had a number of legacy files stored outside the WP structure). Its website looks outdated, but nevertheless, in my experience, FreeFind currently offers one of the best options for site search at a (still) affordable price, especially after some competitors closed doors. (And support is helpful.) FreeFind also now offers an overlay option, that allows search results to be superposed on the page instead of leaving it: I like that!