Apple’s default browser Safari is known for its simplicity of usage. In the same vein, it’s super easy to apply settings in Safari that allows you to open a new window in private mode by default.
The best part is that it is literally two-step process and does not require you to create a script like the other browsers. Here’s what you need to do.
Launch Safari and then go to "Settings". You can use the Command + , keyboard shortcut, or find it in Safari > Settings.
Make sure you are in "General Tab", then look under “Safari opens with:” and select “A new private windows” from the dropdown.
From now on, whenever you launch Safari, it should open the browser in Private Mode by default.