My situation isn't quite as bad as yours tarbat as I usually have about 40-50% of flights displaying flight ID's.
I could do as you've done and move the Company field over but I find it takes up to much space as I like to have the altitude, heading, vrate etc. dislayed. That way I can 'rough guess' the location of many flights that aren't showing global location when combined with the ATC transmissions I'm scanning.
Having the logos displayed would save me having to click on each flight that I don't recognise the registraton of in order to find out which airline it belongs to.