(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.)
I did a reinstall of windows in November, installed Brave and synced the settings including the site searches. In normal (non private) windows the site search always worked for me, but not in private windows.
Investigations:
When I found this topic I started testing a little bit myself and found some interesting issues. Editing or adding site searches always works directly in the normal window, but
Editing the site search seems to work in private windows after opening a new one. But not always, a colon seems to work, but I had problems with just text for the shortcut.
When adding a new site search I had to restart the browser to get it working in a private window.
tl:dr
I guess there are problems after a sync with site searches and generally for private windows depending on the workflow.