Call of Duty Warzone is not listed as a game supported by automatic profile switching. Likewise, the game run time and stats are not tracked in the games playtime section.
The default installation path for the game and executable to track are listed below.
c:\program files (x86)\Call of Duty Modern Warfare\ModernWarfare.exe
Guest
Curated games list has been removed due to removal of the Games panel. For profile syncing on program launch, any .exe can be added in Settings -> Profile Sync.