SUMMARY For my default browser I have set a shell script that calls `firefox --private-window $1` which lets me open all links and searches in a private window. It would be useful to include this functionality in the settings with a checkbox that can set the default browser to use private/incognito mode.
The way that each browser implements private browsing is different, so I'm afraid there is no generic way for us to do this. What each browser's developers (or their distro packagers) could do however is ship multiple desktop files with each browser: one would open in private browsing mode, and the other would not. This would populate the browser list with those options, and you could select that one. For that matter, you can accomplish this yourself by creating an appropriate desktop file, or modifying the browser's existing desktop file. You can use the kmenuedit app for this. But there's no way for Plasma to do it for you, sorry. :)
I've already created a custom desktop file for private mode, it seems to behave glitchy, for example if regular Firefox is pinned opening the private version will be treated as a separate application. Using a shell script works well though. What I was thinking was if Firefox or a popular chromium based browser is default the `--private-window` or `-incognito` flag could be set and if the browser is unsupported the option would be hidden. Since there are only a few popular browsers would this be possible?