Braus is new to me, tho at a glance looks less capable overall than Junction.

Looks like Junction supports firefox profiles via .desktop entries, so it would also support launching via the private CLI flag (firefox -private-window iirc).

Might be easier to keep up with than modding Braus source code in the long run.

https://github.com/sonnyp/Junction

Like other gnome circle apps it heavily encourages flatpak but its available on arch official and other distro package managers.

On android I use URL Check (https://triangularapps.blogspot.com/search/label/UrlChecker?m=1) which has a lot of utility for unshortening snd dropping parameters as well as opening in a specific app. It can open private or normal, too. It is a bit overkill if you only use 1 browser, but tbh, its super useful for revising links before you open them anyway to remove tracking garbage.

I’m not aware of anything this involved and able to do private vs non-private on Linux or macOS, but it would be neat if Browserosourus (macOS) or Junction (Linux) could add those capabilities.

I haven’t looked too much for solutions outside of mobile. Easy enough on desktop to copy a link and paste in a private window. Not as easy on mobile.