But you don't need a status icon to run in the background.
If Firefox wanted to, they could make Firefox continue running in the background. They could even app a system tray entry for Firefox to access recently visited sites or favorite sites, like what Steam does.
This paradigm is actually the norm on MacOS. When you X out of an app, it doesn't actually close. It will just have no open windows but stay open on your dock.
Updated the title