I know it’s only a half-measure compared to your request, but you can long-press the app icon for Brave browser, then long-press = next to “New Private tab”, drop that shortcut where convenient and use it instead of the regular app icon.
Also if you encrypt your device and set a reasonably strong password (not a pin, pattern, etc.), then not deleting history should be somewhat less of a concern… depending on the risks you have in mind. (Maybe write down the password & keep it somewhere safe, just in case.)
You can also temporarily put your device in Airplane mode to see where links from other apps lead to without visiting them… then, copy&paste into Private tab if needed.