This is kinda an inbetween a bug report and a suggestion:
Maybe you should change the wording for the Browser Privacy Mode setting.
Atm it does say "Hides searches in your browser's history by using POST requests. Query will not be shown in page title and you won't be able to go 'back' to a previous search."
But that's not 100% accurate in all cases.
While it does work if you use the Kagi page directly it doesn't hold true for when you input the query into the browser address bar.
At least in Firefox queries you type into the address bar still show up in the browser history. (I assume it just grabs the URL before getting redirected to the post version)
Maybe change it to something like "Hides searches in your browser's history by using POST requests. Query will not be shown in page title and you won't be able to go 'back' to a previous search. Note: This only applies to search queries that are directly executed through the kagi.com search form" or something like that.