Description of the issue:
How can this issue be reproduced?
Create site search in brave://settings/searchEngines
Save it, test it on non private windows it is working
Test it on Private windows it is not working!
Expected result: It should also work on private window
Brave Version( check About Brave): Brave 1.84.141
Additional Information:
Try typing :yt instead (then hit space after the t).
that’s different with features in non private window 
But I have checked today, its now working.
The method I used above works in both Private and standard browsing windows.
Just checked here, and it’s working as you had described (and as @Mattches described, both with and without a colon).
Though I see it’s now working for you as well per 3 days ago.
(I personally added a “gw” in my search to bypass Google’s “AI” results and perform specifically a “Web” search, which allows more verbatim text, the “old school” Google search if you will…to an extent Google allows.)
